But if Peter is "Peter" because he is the First for right of birth (he is the first born), hence predestined from the birth to become the First,...
...then a Paulinist would disagree: for a Paulinist, it is Paul who was predestined from the birth (read Galatians 1).
Hence, the our Paulinist would have given to Peter the name of Peter not as his original name, but only as a late surname given by Jesus:
These are the twelve he appointed: Simon (to whom he gave the name Peter),
(Mark 3:16)
...hence eclipsing deliberately the fact that Peter was Peter ('the first born) from the first day.
Debt to Stahl for this view (but note that Stahl has the Fourth Gospel preceding Mark).