Does the engine alternator charge the house batteries.
Well.. on motor homes it is supposed to.
How to tell if it does (The easy way)
Unplug. Turn on an inside light. .Note illilumination level
Start engine.. it may take a minute or two but does light suddenly get brighter? IF SO then YES. That simple no test gear needed.
IF Light does NOT get brighter.. Two posibilities
1 Some LED's won't (Option is the fantastic fab or max air fab will run faster)
Second if nothing changes within 3 minutes of starting engine.. PROBLEM exists seek help from someone familiar with shooting tribbles.
(Believe it or not I had to tell David Gerrold how he "Took a dump" on James T Kirk. he'd forgotten).