Definitely not 10 minutes, getting the back off was insane. Poked and pried for an hour or so before I finally got it to come free in one spot, then worked a guitar pick tool around from there to finally get it completely open. Once open, it was pretty easy to get the battery tray out, but the only replacement battery I could find didn’t come with a tray, so I spent another large chunk of time wedging the flat end of a spudger between the tape and the old battery and rotating it back and forth to peel a bit of battery off the tape at a time (the old battery had inflated pretty good, and fortunately I didn’t pop it). Went back together trivially, seems to work, and is charging now. The screen is flat again, not bulged in the middle.

Did it, worked like a charm, none of the tape was too sticky to get things out. My old battery was “pregnant” swelling the case. Fortunately I got it out without any leaking of whatever nasty stuff is under the film the battery is wrapped in. Top priority now is to get it to the recycling center before it does anything like burst into flame :-).