Hi S and S,
Actually axolotls can live in the fridge for months on end, there are stories of people who keep them in the fridge permanently, so fridiging itself isn't dangerous per se.
The only drawback is the reduced appetite due to the reduced metabolism. If your axie appears completely healthy and fine and your main tank is all sorted out, i don't see why you can't move it back home.
Just remember to let the water equilibrate to ambient temperature before you put your axie back in the main tank.
Provided that the tank parameters are fine, your axie's appetite should pick up.
Cheers.