If you DRAIN the battery completely, there is no energy to start a fire.(I don’t suggest throwing it in a fire though to check. :person_facepalming: ) I have not heard of any problem of this sort associated with them.
Charging is a larger potential risk. Then you are putting energy IN. ANY Lion with protection (or not) plus charger that you rely on without any oversight at all has a similar risk associated with using it.
Note - the amount of lithium in a AAA is……tiny……Probably 1tsp of gasoline is more dangerous.

Size should be fine, but I haven’t used this brand. All the other ones I have seem well made and don’t have a fit problem.

FWIW I have a bunch of 9v (2S) PowerUS batteries that have been used HARD for over 10 years and are still working fine. They use a similar circuit. I’ve had one fail. No danger, it became a ‘dud’ and would no longer charge.
I also have a few 9v EBL. They simply aren’t as good (worse capacity and losing it faster), but I don’t consider them more dangerous.