Had a similar/exact issue with my Red AXS rear derailleur. One day it just stopped working. If I pulled the battery for a while, best I got was 10 seconds of shifts before it died.
What eventually worked was pulling the battery and leaving it sit for a few days. I wrote it off and got a spare RD in the meantime, then by chance figured I’d try and see if it worked. Sure enough, shifted fine … 