Not a solution, as the solution is quite obvious, but a potential glitch:
I beat the first 4 levels with one custom made ship and then beat the 5th level with another custom made ship in a different session, all in Veteran difficulty level, and this trophy did not unlock.
So, make sure to use the same custom ship, and I am guessing it does not need to be done in the same session.
If anyone can verify this info, please do.