The relay mentioned above looks like this
and is $15 on Amazon.
The idea is that the Chassis Engine, when running, sends 12VDC to one of the small terminals. The other small terminal is probably grounded.
That voltage closes the contacts between Chassis Alternator and Coach (House) battery, so it's charged when the Engine is running. The Coach/House cannot discharge the Chassis Battery because they aren't connected with the Engine Off.
Do you have an Assistant? If so:
Open Hood, Start Engine.
One of you put your DVOM across the Coach Battery.
The other of you put a jumper across the two LARGE terminals.
Voltage at Coach Battery goes UP?
If Yes, Test of Relay satisfactory and complete.
If NO, Relay is either not being powered through Chassis (one of the small terminals) OR other small terminal not grounded
OR if NO, bad Relay Coil (between small terminals) or bad Relay Contacts (between the Large terminals). Spend $15. Make it $17 and buy the Assistant a beer.