Which Catholic writer(s) has compared St Paul's "third heaven" to the third stage of interior prayer? There is a Catholic author who compared St Paul's vision of the third heaven to the third or final stage or way of perfection. I can remember reading it some years ago, but cannot recall it now.

This concept is not easy for me to understand. Many authors have written about the Ways of Perfection, But I am looking for the one who used St Paul's vision of 2 Corinthians 12:1-7 as a comparison to the unitive state of prayer. 


  Paul’s Vision and His Thorn
  
  12 I must go on boasting. Although there is nothing to be gained, I will go on to visions and revelations from the Lord. 2 I know a man in Christ who fourteen years ago was caught up to the third heaven. Whether it was in the body or out of the body I do not know—God knows. 3 And I know that this man—whether in the body or apart from the body I do not know, but God knows— 4 was caught up to paradise and heard inexpressible things, things that no one is permitted to tell. 5 I will boast about a man like that, but I will not boast about myself, except about my weaknesses. 6 Even if I should choose to boast, I would not be a fool, because I would be speaking the truth. But I refrain, so no one will think more of me than is warranted by what I do or say, 7 or because of these surpassingly great revelations. Therefore, in order to keep me from becoming conceited, I was given a thorn in my flesh, a messenger of Satan, to torment me.


In Catholic thought the three stages (ways,state) of interior life or prayer are the purgative stage, the illuminative stage and the unitive stage.

This link I hope, can get the idea of where I am coming from: On the Purgative, Illuminative and Unitive Stages of Spiritual Life, as seen in a Cartoon.
Perhaps you are thinking of "The Interior Castle" by St. Teresa of Avila, Doctor of the Catholic Church?  On page 77 of the document at this link, the introduction to the 3rd Chapter, she specifically cites 2 Cor. xii. 2-4.

This sounds to me close to your question, though this is not the final stage of perfection in that book.