Someone over on Reddit seems to have located a "new" version of the Apple //c diagnostics compared to the only version on Asimov. This supports the newer ROM 3/4 versions, as it has an extended memory card test. The existing version on Asimov seems to only like ROM 255 machines and forces you to have a serial loop back cable. This one lets you skip the serial test if you don't have the cable.
Interestingly, I ran this on my rom 4 and found something odd. There is a system status screen in program. It says "current rom version: 06 (original IIc ROM)". But if I do a "print peek(64447)", it does in fact return a 4. So maybe Apple numbered them differently than what people generally go by? My rom 255 is currently packed up. Maybe I'll dig it out tomorrow and see how this reports the rom version on that. It also says "memory card status: not supported with this rom". I do have an a2heaven 1mb expansion card installed. And it did pass the extended memory test. So I wonder why this rom is listed as not being able to tell the status of the card.