Fly with AirTag?

AFAIK, the FAA (not sure about other country's similar agencies) does not prohibit the use of Apple AirTags on either carry-on or checked baggage. Typically, the airlines follow suite. However, if you are concerned, you can always contact the airline to find out their policies around this matter.

well they seem to sell them to help find luggage so one would assume as much

the battery is a std. CR2032 battery used to tons of consumer electronics so if you can't fly with those its few consumer electronic devices you can fly with.