So specifically the issue is with the sound? First thing to do is reflow the V ROM chips and the PCM chip on the PROG board.
If that doesn’t fix it, there could be broken solder pads that have separated from the board but are still soldered to the chip pins, this is most common with surface mount chips, like the PCM chip. But, the description of the cartridge working fine after warming up tells me it’s probably just cracked solder joints.
if all those things check out and the sound still has issues, then you look at replacing the V ROM chips. The quick and dirty way to figure out roughly which ones are broken is to load Pulstar in an emulator but swap out one of the V ROM files with an empty dummy file so the chip behaves as if it’s dead. Do this once with every V ROM file and see if any particular chip matches your symptoms.