Ruptured air sacs is my game! Had one damaged after a hawk attack, poor guy started feeling real puffy. This is subcutaneous emphysema and you can find youtube videos on how to alleviate it.
You are going to need a short small bore needle, hold that little critter down to keep them safe, target the most inflated area or attempt to milk the air into one area, inject ONLY TO PIERCE THE SKIN (this is not an injection), then withdraw air into the syringe and repeat. If you have a large enough needle you may be able to squeeze the air out of your original hole.
Bear in mind you may have to do this repeatedly. I do not mean to alarm you, but subcutaneous emphysema can put excess pressure on organs and the windpipe so best resolve as soon as possible.