Coach - yes. Your convertor-battery charger will take care of these.
There is a good possibility it will not the keep the chassis battery
charged up. Some systems will not charge the coach and chassis from shore power.
There are lots of monitors that the RV factory slips in and pulls from
the chassis battery. These will pull it down in a couple of weeks.
I do the same-thing you are talking about and added a a small battery maintainer
that I bought from Walmart. It plugs into the dash power port to charge the chassis battery and I plug into the 120V AC inside the MH to power it.
Not all batteries stay charged.