I'm trying to get this mirror out without making a total mess of the surrounding drywall. Any best practices?
3 possibilities that I can see.
Tape every square inch of it and just start smashing. Wear serious eye protection and a face shield, gauntlet gloves, heavy clothing, head covering, ... in short, don't have any exposed skin.
Tape, protect yourself (as above), and try to get a pry bar under the edges. You'll need to do a bit of drywall repair, but you might get lucky and not have a pile of shards.
Buy (rent?) a specialty suction cup slide puller. (This is scary at the best of times and probably unavailable at normal rental...) Edit to add youtube link: https://www.youtube.com/watch?v=dQne8Caw1LI