Many coaches do charge the chassis battery on shore power. Mine was equipped with a B.I.R.D. If your question is just for your MH plug it into shore power and check the voltage at the coach batteries and then the chassis battery. If they are the same, turn on the parking lights and read chassis voltage. If still the same, then yes yours does. If the volts drop on the chassis battery and stay down then no it doesn't. Digital volt meters are available for under $20.00 if you don't have one. I saw one at Harbor Freight for under $10.00.
2016 Wildwood 28RLDS
2014 Tundra 1794 Platnum,"23" T-Bucket
2011 Harley Softail, 2010 Spyder RTS