The inverter has no control over what it charges. Only how it charges.
IF the body builder used a BIRD type battery isolator (Bidirectional Isolator Relay Device) then both engine and house chargers charge each other's batteries.
If not. then not.. However there is an add on (Trickle start) that will add the reverse capability.
How to find out? Your volt meter is your friend.
Park, shut off engine and plug into shore.
Set up. take your time. or even over night
Take voltmeter or multimeter set to DV volts 15 or 20 volt range (Depending on meter) Measure voltage on house batteries AT THE BATTERY.. Should be over 13
Measure chassis (engine) over 13 being charged (Float voltage) under 13 NOT. (General guideline.. specific voltage depends on thigs like Chassis should not be higher than house battery when shore charging) (Overnight means house SHOULD be full. )
Home was where I park it. but alas the.
2005 Damon Intruder 377 Alas declared a total loss
after a semi "nicked" it. Still have the radios
Kenwood TS-2000, ICOM ID-5100, ID-51A+2, ID-880 REF030C most times