The ROM makes no difference. The result is the same with either one. How would one go about replacing U16 and 17? These appear to be soldered directly to the board. Also, where would I obtain these replacement parts? I imagine I'd have to order them from somewhere on the other side of the pond since I'm in the US?
Pretty much everything on the A600 is soldered apart from the ROM. ROHS standards came in 1992. Lead free solder, so count your blessings that fixing it isn't going to cause health damage.
Spare parts you are looking for are an 8520 (CIA, Complex Interface Adaptor) in PLCC or is it FPGA form (? square SMD) for U7, and if that doesn't work, the RAM chip or Chips. I don't know that off the top of my head, but it's written on the chips.
You may or may not be able to get spare parts. From anywhere. But, they're generally to be had, if you look hard enough and are patient.
As for the soldering skills, if you don't have, find someone local that does. Makerspace or Hackerspace close to you? Fablab, anything like that? Worth a search engine check on your location.
Solder containing lead is now pretty rare, but still available. No more is going to be produced though - at least, not in the USA, Europe, or Japan.I think China has severely restricted the use of lead, but the Chinese economy is so huge, it's very difficult to be sure that environmental monitoring is checking up on that, as far as solder goes.