Could you generate a 12-flip with a 6-flip plus a 2-2-2-2-4 cycle pattern? E.g. [(R' F R U)5, D2 F2 R D R' U' B2 R2 F' U' F R2 F2 D' U F2 U2] (note that that last sequence performs a corner 2-cycle, so we can easily verify that the edge permutation is odd).
[U2 L F2 D2 F2 U2 L2 B R B' L' U' R' D2 U' L' F2 L2 U2, U2 R2 U2 R2 B D L2 D2 B2 U' B R' U' L' U F' L2 R]
It seems like having a 4-cycle as a part of my z is okay if I am just doing pure orientation with the piece type which has that 4-cycle. The pure orientation might be the only exception which allows z to be a cycle class other than a product of disjoint 2-cycles, but then again, this is the only case so far that appeared to not being able to be reached with the restriction of just having a product of disjoint 2-cycles. I'll have to look into this more to see if the parity of two different piece types can have any other potential conflicts. If not, then it will be a question of whether or not all corner orientations and edge orientations can be solved with one commutator (using my method) separately. If so, then they can unquestionably can be merged.
I have almost finished proving that 3-cycles of corners + any corner orientation case can be solved with one commutator, but that's as far as I've gotten because it's very difficult work. This is going to be a brute force search, and I don't have a program to do this for me, so it might take a while to go through all of the cases by hand. The amount of tests for cycle classes which involve more pieces will increase as well, which doesn't help. (I might need to do around 4000 tests just for corner orientations).
Thanks a bunch! I'm really hoping that the nxnxn cube can be solved with one commutator. I guess we will see.