With out spending a lot of time and money, to find out if your chassis batteries are hooked up to the house charging system is to get one of these
12 Volt Battery/Alternator Tester
If the alternator side shows that you are good, then the chassis battery is also being charged when you are hooked up to shore power.
If not, then a $10.00 battery tender hooked to the chassis battery and plugged into shore power for maintaining the battery through the winter months is all I use.
Automatic Battery Charger - 12V
I cut the alligator clips and directly hook the "cheap" battery tenders from harbor freight directly onto the batteries in my back hoe, tractor, mower, basically any piece of equipment than might not be used for long periods during the winter. They keep the battery charged, and help keep it from freezing. They have been amazingly shock and weather resistant for years on the back hoe.
I'm not sure if I've answered your question exactly, but I hope it helps.