Thanks for the ideas.
The batteries I had did have a bitterant coating. Based on other posts, I wiped them with isopropyl alcohol. As the coating is clear, I don't know if I removed it, or not.
However, I did try permutations of the 5-press reset procedure, as well, with a couple of different batteries.
The net effect was that I killed two batteries. The AirTag would sometimes register, when it was first reset, but it wasn't a persistent behaviour. I saw it register twice, in all my experiments.
I bought another battery (Panasonic). My battery meter says it's good. However, now my AirTag won't even chirp, when the battery is seated into it.
I'm assuming it's dead, and that it was short-circuiting my batteries on the way down. Any idea how long these things last and what finally dies within them?