No drive swap's needed: managed to fix it using Byrd's link. FYI It's the 12x model.

I am not interested in a replacement battery unit, since I can't imagine something already 13 years old being reliable for very long. I'd rather rebuild the dead one I have.

So far the working battery seems to be getting progressively better, from an uptime of just 5 seconds to almost an hour now, and counting.