"The battery/power supply alone is going to be at least an order of magnitude larger in volume."

Indeed, people do use rechargeable lithium batteries like the 50mAh CR322 (22mm by 3mm) or smaller CR311, but these would only be able to handle a 11mA EFR32BG29 output load for a few hours at most... Yet in a mostly dormant power save mode YMMV =3