I could not have done it without YouTube. Luckily a search finds a video which even I did not find irritating. Somehow he moves at a quick enough pace that you can easily stop-motion at the parts you need.
I could not figure out how to unlatch the battery, and eventually, after seeing what he did, used a sewing needle to do it.
For a while I thought the watch might have just plain died, but eventually I managed to accomplish the "short to reset".
I disassembled it more than I really should have, but it seems properly back together now. All five buttons even work easily (other Timex I've had have had some switches get really difficult to use). I might have mucked up the seal too, but we'll see. I'll use the replacement watch from now on, unless perhaps I encounter a particularly warm sunny day and will use the old one experimentally.
Or if the new one dies, I guess.