Based on what verse did Luther know that no sin can separate the believers from God? The article in this link said:


  Something has gone wrong in Lutheranism. It has been more than
  adequately shown that regarding the doctrine of the eternal
  predestination of God Martin Luther taught things directly contrary to
  the standards of Modern Lutheranism


Which to be honest, I still cannot understand that "the Modern Lutheran Church has gone wrong" is in a positive sense or a negative sense in the view of the writer in that article. 

In other words, it's not clear to me whether the views of Modern Lutheran Church is an apostasy or not in the article's writer point of view.



Anyway, Luther said in this link:


  No sin can separate us from Him (Martin Luther: Letter to Melanchthon)


One of my commentator said:


  Luther knew that no sin is 'too great' to separate us from the love
  and forgiveness of God, in Christ


So the question is:
Based on what verse did Luther knows that no sin can separate the believers from God ?
My understanding is that the Lutheran Church believes it is possible for someone to lose their salvation because of unrepentant and persistent sin. I doubt they would interpret Romans 8:38-39 in the manner you suggest.  Here is some information I found that may be relevant:


  At times, our pastors, with considerable sorrow, may have to tell a person not to attend the Lord’s Supper until he or she has repented of sin. If the situation continues without repentance, the pastor may have to declare to the person, on behalf of the congregation that has made this decision, that he or she is excluded from the church until there is repentance.  Excommunication is a last resort to help a person recognize the extremely dangerous situation he has placed himself in because he will not repent of his sin. It is a final attempt to win someone back from Satan’s influence.   Source: https://www.faithlutherancorning.org/confession-absolution


The Lutheran Church believes that repeated sin hardens the heart and may lead to sinning against the Holy Spirit.  Also, as noted in the previous extract above, unrepentant and repeated sin could result in being excommunicated:


  SIN: Transgression of God's law (Ro 4:15; 1 Jn 3:4). Sin may be divided into original sin (see Sin, Original) and actual sin. Actual sin (every act, thought, emotion (e.g., lust*) conflicting with God's law) may be involuntary or may be done ignorantly (Acts 17:30) and includes sins of commission (cf., e.g., Mt 15:19; Ja 1:15) and sins of omission (Ja 4:17). Sin arouses God's righteous wrath and deserves His punishment. Willful sin sears conscience*; repeated, it hardens the heart; may lead to, but is not identical with, the unpardonable sin against the Holy Spirit.’  http://cyclopedia.lcms.org/display.asp?t1=s&amp;word=SIN


This extract explains the Lutheran view of sin, but it does not suggest a believer can persist in unrepentant sin and "get away with it":


  VENIAL AND MORTAL SINS: The Lutheran Confessions speak of sin* that is mortal, or deadly, i. e., irreconcilable with faith (Ap IV 48, 64, 109, 115). When believers fall into open sin, faith has departed (SA-III III 43–44). One who obeys his lusts does not retain faith (Ap IV 144). Original sin (see Sin, Original) is mortal; it brings eternal death on those who are not born again (AC II 2 Lat.). One who is dead in sin is insensitive to sin (LC, V: The Sacrament of the Altar, 77–78). Sins remain in believers (SA-III III 40; FC SD II 34). Many regard the following as 7 deadly sins, fatal to spiritual progress: pride,* covetousness,* lust,* anger, gluttony, envy,* sloth. But man cannot weigh, distinguish, or differentiate sins; all sins manifest total corruption (SA-III III 36–38), merit God's wrath (Mt 5:18–19; Gl 3:10; Ja 2:10), and are deadly (Eze 18:4; Ro 6:23); every sin loses its deadly effect when Christ, apprehended by faith, intervenes (Ro 8:l; 1 Jn 1:7, 9; 2:1–2). Source: http://cyclopedia.lcms.org/display.asp?t1=s&amp;word=SINS.VENIALANDMORTAL


I am not a Lutheran and I may have misunderstood what you are asking (because it is not clear).  

Edit re your view that Martin Luther believed it was all right to continue in sin:  Wrong. Martin Luther nowhere supports that interpretation of Romans 8:38-39. That partial quote from the letter Luther wrote to Melanchthon has been taken out of context.  Please note: “The Modern Lutheran church does not stand with Martin Luther on the issue of predestination, and thus suffers from an internal contradiction. Its efforts to modify Luther's views and to present a more moderate case for predestination ultimately end in conflict with Luther's uncompromising doctrine of God's Sovereignty.”  https://www.monergism.com/thethreshold/articles/onsite/double_luther.html