this is probably similar to the way pinball machines work.
there is a chip that is on the cards that facilitates the connection of the roms to the main board CPU. it allows the CPU to share the dataline with multiple roms. likely one of the chips is bad causing neither of the cards to properly wait for their turn to be accessed. probably a 74LS300 series chip. if you have the ability to obtain the parts on the board that aren't the main roms, give them a swap.
i don't know 100% as ive never tried to fix one of these machines but this sounds like an issue with corruption you can get with interface chips in pinball machines (sound and video).