bigmessowires sent me a second ROMinator containing the stock IIci ROM image (thanks!) to experiment with. On the mostly-working board, it boots just as well as with the onboard ROMs. From talking to bigmessowires a bit, he noted that the way the ROMinator custom image may still be exercising the hardware in ways that the standard ROM doesn't, so there may be lurking problems still on the new board that I have to hunt down. Going back to the battery-damaged board, I get silence booting from the stock IIci ROMinator, even after replacing the original VIA with a fresh W65C22N, so there's still some issue on that board that's causing it to get hung up before it can chime.