How does one change the battery in the Air Tag?

No! Do NOT pry it. That will only potentially damage the tag. Instead, you just need to press down on the back (silver) cover with two fingers, and then rotate it. IdrisSeabright has already provided you with the Apple Support link on how to do this properly.

Should the cover not come off easily, it may be possible that the battery inside is swollen, and it may take a bit more of a twisting force to open it,

I don't think prying should be involved.

How to replace the battery in your AirTag - Apple Support