@Anna C
One option to consider is a sort of compromise between the BP and MP if you want to just do one lift per PTTP, while remaining true to the principles and programming of PTTP.
My understanding of the program was that Pavel put the side press in for convenience as a person could use the same bar they were deadlifting with, needing no extra equipment, nor a bench. But the underlying principle was to have a pressing movement to complement the pull and lower body work of the DL.
I do the incline BP for my press, as it's working the shoulders more and keeping you strong with the BP. It should help both your MP and BP.
I'm currently in a cycle of just DL and incline BP for my lifts with no extra mayo.
But I just hit a new personal best with my KB overhead press, despite not training it. I just thought I'd clean and press the 32 kgs bell for fun after a session and see what happened.
This is, I believe, the "what the hell' effect." I damn near dropped the bell in delight when I pressed two bell sizes more than I'd previously pressed.
There's something magical about PTTP.
Therefore, I chose the incline BP as the press, and it's working out very well indeed. You might consider it.