Apart from any claim related to antiquity, does the modern Roman Catholic Church claim to be ‘The Body of Christ'? An argument to be the Universal Church (made by any organisation) might be the argument of continuity and antiquity, that an organisation can trace its existence back to antiquity such that it can trace a continuous presence in the world back to the time of the Apostles.

Laying that argument aside, does the present day Church of Rome have any claim or argument to make that it can be genuinely called 'The Body of Christ'?
Apart from any claim related to antiquity, does the modern Roman Catholic Church claim to be" The Body of Christ"?

The short answer is: Yes.

Pope Pius XII's encyclical Mystici Corporis Christi (Mystical Body of Christ) (June 29, 1943) should count as being on topic for this post? And, if so, keep reading.

Although Pope Pius XII states in his encyclical states that the true Church is which is the One, Holy, Catholic, Apostolic and Roman Church as being the Mystical body of Christ; he also states that the Universal Church comprises of the Latin and Eastern Rites.


  
  If we would define and describe this true Church of Jesus Christ - which is the One, Holy, Catholic, Apostolic and Roman Church - we shall find nothing more noble, more sublime, or more divine than the expression "the Mystical Body of Christ" - an expression which springs from and is, as it were, the fair flowering of the repeated teaching of the Sacred Scriptures and the Holy Fathers.
  
  
  (42) What we have thus far said of the Universal Church must be understood also of the individual Christian communities, whether Oriental or Latin, which go to makeup the one Catholic Church. For they, too, are ruled by Jesus Christ through the voice of their respective Bishops.


Of all the Eastern Rites Churches, the Maronite Catholic Church has never been separated from unity with the See of St. Peter. Thus indicating a small flaw in the title of this question as it is presented at this moment in time.


  The Maronite Church is considered the only one of the Eastern Catholic Churches to have always remained in full communion with the Holy See, while most of the other churches unified from the 16th century onwards. 


Before going on any further, I believe the word "Roman" should be used with caution. The reason being is that the first three centuries the liturgical language in Rome was Greek, not Latin. Some believe that it was St. Augustine who first brought in Latin in to the liturgy while in Africa. The Church has from her very foundation worshiped in several languages and still does. This is not an historical claim, but an historical fact and as such is permitted in this response.

The following is not an historical claim from antiquity, but an historical fact dating back to antiquity. 


  The first language of Christian liturgy was Aramaic, the common language of the first Christians, who were Palestinian Jews. While Hebrew was the language of scripture and formal worship, Christian worship occurred in the home where Aramaic was spoken. The words Abba and maranatha are Aramaic.
  
  Christianity quickly spread from Palestine to the rest of the world, and the Eucharist came to be celebrated in many languages, including Syriac, Coptic, and Armenian. In most of the Mediterranean world, the common language was Greek, which became the language of liturgy in that region and remained so until the early third century.
  
  Eucharist itself is a Greek word, meaning thanksgiving. The phrase Kyrie eleison and the words liturgy, baptism, evangelize, martyr, and catechumen, among other familiar church words, are also Greek in origin.
  
  From around the third century B.C., what we call “classical” Latin was the language of the Roman aristocracy and the educated classes. Around the time Jesus was born, during the reign of Augustus Caesar, the language began to change. The Roman aristocracy was destroyed by war and political infighting; when they disappeared, their language went with them. Classical Latin was replaced by a less refined version of the language.
  
  In the third and fourth centuries A.D. this form of Latin began to replace Greek as the common language of the Roman world and soon became the language of the liturgy.
  
  Exactly how this change in the liturgy came about is uncertain. In the early church the liturgy was led extemporaneously by the bishop, according to a pattern. There were written examples of Eucharistic Prayers, but they were models, not prescribed prayers. The last such document in Greek was written around the year 215. By the sixth century, the Roman Canon (which is still in use, also called Eucharistic Prayer I) appears, completely in Latin and prescribed for use exactly as written. - When did we start celebrating Mass in Latin?


Now for responding to the question at hand: Apart from any claim related to antiquity, does the modern Roman Catholic Church claim to be "The Body of Christ"?

Yes.

Well it so happens that Pope Pius XII in his encyclical Mystici Corporis Christi (29 June 1943) affirms that the Catholic Church is in fact the Mystical Body of Christ. The pope is the head of the Catholic Church on earth whether of the Roman Rite or Eastern Rite Catholics. We are still one Church in Christ. 


  The Maronite Church is considered the only one of the Eastern Catholic Churches to have always remained in full communion with the Holy See, while most of the other churches unified from the 16th century onwards. However, the Melkite Catholic Church and the Italo-Albanian Greek Catholic Church also claim perpetual communion. Eastern Catholic Churches (Wikipedia)


Pope Pius XII boldly asserts that the Universal Church is made up of the Roman Rite and those of the Eastern Rites united to the See of St. Peter. 


  Mystici corporis Christi (29 June 1943) is a papal encyclical issued by Pope Pius XII during World War II, on the Church as the Mystical Body of Christ.2 It is one of the more important encyclicals of Pope Pius XII, because of its topic, the Church, and because its Church concept was fully included in Lumen gentium but also strongly debated during and after Vatican II. The Church is called body, because it is a living entity; it is called the body of Christ, because Christ is its Head and Founder; it is called mystical body, because it is neither a purely physical nor a purely spiritual unity, but supernatural.


The actual encyclical has much to say and here follows just a small sampling. One may peruse the entire encyclical at one's leisure (Mystici Corporis Christi):


  The doctrine of the Mystical Body of Christ, which is the Church, was first taught us by the Redeemer Himself. Illustrating as it does the great and inestimable privilege of our intimate union with so exalted a Head, this doctrine by its sublime dignity invites all those who are drawn by the Holy Spirit to study it, and gives them, in the truths of which it proposes to the mind, a strong incentive to the performance of such good works as are conformable to its teaching.
  
  (14) That the Church is a body is frequently asserted in the Sacred Scriptures. "Christ," says the Apostle, "is the Head of the Body of the Church." If the Church is a body, it must be an unbroken unity, according to those words of Paul: "Though many we are one body in Christ." But it is not enough that the Body of the Church should be an unbroken unity; it must also be something definite and perceptible to the senses as Our predecessor of happy memory, Leo XIII, in his Encyclical Satis Cognitum asserts: "the Church is visible because she is a body. Hence they err in a matter of divine truth, who imagine the Church to be invisible, intangible, a something merely "pneumatological" as they say, by which many Christian communities, though they differ from each other in their profession of faith, are untied by an invisible bond.
  
  (15) But a body calls also for a multiplicity of members, which are linked together in such a way as to help one another. And as in the body when one member suffers, all the other members share its pain, and the healthy members come to the assistance of the ailing, so in the Church the individual members do not live for themselves alone, but also help their fellows, and all work in mutual collaboration for the common comfort and for the more perfect building up of the whole Body.
  
  (22) Actually only those are to be included as members of the Church who have been baptized and profess the true faith, and who have not been so unfortunate as to separate themselves from the unity of the Body, or been excluded by legitimate authority for grave faults committed. "For in one spirit" says the Apostle, "were we all baptized into one Body, whether Jews or Gentiles, whether bond or free." As therefore in the true Christian community there is only one Body, one Spirit, one Lord, and one Baptism, so there can be only one faith. And therefore, if a man refuse to hear the Church, let him be considered - so the Lord commands - as a heathen and a publican. It follows that those who are divided in faith or government cannot be living in the unity of such a Body, nor can they be living the life of its one Divine Spirit.
  
  (34) That this Mystical Body which is the Church should be called Christ's is proved in the second place from the fact that He must be universally acknowledged as its actual Head. "He," as St. Paul says, "is the Head of the Body, the Church." He is the Head from whom the whole body perfectly organized, "groweth and maketh increase unto the edifying of itself." 
  
  (42) What we have thus far said of the Universal Church must be understood also of the individual Christian communities, whether Oriental or Latin, which go to makeup the one Catholic Church. For they, too, are ruled by Jesus Christ through the voice of their respective Bishops. Consequently, Bishops must be considered as the more illustrious members of the Universal Church, for they are united by a very special bond to the divine Head of the whole Body and so are rightly called "principal parts of the members of the Lord;" moreover, as far as his own diocese is concerned, each one as a true Shepherd feeds the flock entrusted to him and rules it in the name of Christ. Yet in exercising this office they are not altogether independent, but are subordinate to the lawful authority of the Roman Pontiff, although enjoying the ordinary power of jurisdiction which they receive directly from the same Supreme Pontiff. Therefore, Bishops should be revered by the faithful as divinely appointed successors of the Apostles, and to them, even more than to the highest civil authorities should be applied the words: "Touch not my anointed one!" For Bishops have been anointed with the chrism of the Holy Spirit.
  
  (66) And if at times there appears in the Church something that indicates the weakness of our human nature, it should not be attributed to her juridical constitution, but rather to that regrettable inclination to evil found in each individual, which its Divine Founder permits even at times in the most exalted members of His Mystical Body, for the purpose of testing the virtue of the Shepherds no less than of the flocks, and that all may increase the merit of their Christian faith. For, as We said above, Christ did not wish to exclude sinners from His Church; hence if some of her members are suffering from spiritual maladies, that is no reason why we should lessen our love for the Church, but rather a reason why we should increase our devotion to her members. Certainly the loving Mother is spotless in the Sacraments by which she gives birth to and nourishes her children; in the faith which she has always preserved inviolate; in her sacred laws imposed on all; in the evangelical counsels which she recommends; in those heavenly gifts and extraordinary grace through which with inexhaustible fecundity she generates hosts of martyrs, virgins and confessors. But it cannot be laid to her charge if some members fall, weak or wounded. In their name she prays to God daily: "Forgive us our trespasses;" and with the brave heart of a mother she applies herself at once to the work of nursing them back to spiritual health. When, therefore, we call the Body of Jesus Christ "mystical," the very meaning of the word conveys a solemn warning. It is a warning that echoes in these words of St. Leo: "Recognize, O Christian, your dignity, and being made a sharer of the divine nature go not back to your former worthlessness along the way of unseemly conduct. Keep in mind of what Head and of what Body you are a member."


Then there is the fact that the Church claims of Apostolic Succession; but because this question overtly refuses to allow such legitimate claims (in Antiquity) shows yet another flaw on the subject (wording in the title of this post) of the true Universal Body of Christ which is the Church.

The Pope is the Supreme Pontiff of the entire Catholic Church (Latin or Oriental) which is called the Mystical Body of Christ, yesterday and today.