On a 2006 there is a good chance your isolator is a BIRD (Bi Directional Isolator Relay Device) and the house converter does indeed charge the chassis.
I'd also check the chassis battery with a good load tester.
Suspects on a don't start
Bad battery
Dirty battery connections (See note below)
Bad pilot relay or solenoid or starter
Bad other connection
My expierence
1st time (2nd and 3rd and perhaps 4th) it was a dirty battery connection. Body builder made that cable and used the WRONG terminal material. The next failure was a bad battery.. When I replaced the battery I replaced the terminal so that has not bothered me again (Been years)
The last failure was the pilot relay. How to tell? (I did it from the drivers seat)
The Pilot relay makes a sharp "Click" sound.. Like the relay on your Air Conditioner or other relay operated device. (Very soprano)
The solenoid (on the starter in my case) is a much more solid CLUNK (more Bass like).
Of course chatter is low-bat/bad connection.
I cleaned the pilot relay and VAROOM