Though a pain to do, try removing the battery when not in use, it'll help determine if the issue is with the RD or battery.
As for 3rd party, I've used a 3rd party battery (RSR brand) a few times without issue on an 11s and AXS.
SRAM 11spd etap derailleur battery
Moderator: robbosmans
2021 Giant TCR Advanced SL0 Disc 6.8kg
2019 Specialized Allez Sprint Disc 7.9kg
2017 Festka Scalatore 5.375kg
Original account [shotgun]: memberlist.php?mode=viewprofile&u=10594
2019 Specialized Allez Sprint Disc 7.9kg
2017 Festka Scalatore 5.375kg
Original account [shotgun]: memberlist.php?mode=viewprofile&u=10594