The answer should be: it is impossible.
Just make a simple move and you will have something that you can solve with commutators.
Parity only can happen in even frames 2x2, 4x4, ... And if you want to solve it blind, you could check for parity first and if there is parity then trace with one move done (and don't forget to apply that move when you start the exe

).
But on a 4x4 it should be easier to fix it at the end:
with buffer B and target A the first row will be BACD, shift it 1 to the left to get ACDB and then solve the CDB with a commutator.