Followed recommendation of ATSCNTSC. Purchased 2-pack from Amazon at $6.99 -$2 more than a single).

Lift battery pack from housing. Brake tabs/wires off the battery. Cut old heat shrink tubing off. With soldering gun, heat & pull tabs off those wires. Tin those wires. Remove tape on top off new battery. Make two wires come down middle of batteries instead of out the side. Put tape back in place. Cut connector off new battery pack appx 1 1/4" away from the battery. Tin those wires. Slide on a short section of heat shrink tubing. Prop circuit board against something. Reconnect / solder battery wires together. slide heat shrink to cover. Fold the newly soldered/heat shrinked wire down into the black plastic housing. The longer 1 1/4" wire allows a back and forth fold. Continue re-assembly.

Revived OLD BG2040. Works great!

Kastar 2-Pack AAAX2 2.4V MSM 1000mAh Ni-MH Rechargeable Battery for Uniden BT-1008 BT-1016 BT1008S DECT20602 DECT 2080 DECT 2060-2 DWX-207 DECT20854WX DECT21802 Dantona BATT-1008 Empire CPH-515B

Thank you for taking the time to write this up and provide link to the pins! Definitely a weak point in the home Ember mug2