Battery disconnect - can I still use my heat?

Yes, assuming your converter is roughly ten years old or less.

YEP^^^^^

If you are going to be plugged into AC power source then leave batteries in place, disconnect on and converter on.....
Converter will supply the DC System AND maintain the batteries with float charge

IF you remove batteries....disconnect off and leave converter on for DC Power....furnace runs on DC

Since most power centers are also a battery charger, I'd just leave the batteries connected. It will not shorten the life of your battery unless there's a problem in the charging circuit of your power center.

I will say you can. Everything in my TT works on shore power, with the battery unhooked. I wouldn't worry about the converter. I can't turn mine off.