Does it work while started or in Acc position ? I'm not sure how it would be wired to only work while running and not in run position.BUT...
My radio has some sort of relays that will swap radio "and cig-lighter I believe" to house batteries once the key is off or chassis battery is disconnected. Looks like a PO added a switch to choose which battery also maybe it's OEM but switch is not.
95 Monaco Crown Royale
M11 400hp, 4060 trans.
Aquahot, Generac Guardian7.5k