what battery is used for starting a generator in a class C

My Chevy Chassis RV also starts the generator from the Coach batteries.

Your idea of checking the batteries with a meter is a sound idea, those lights in the control panel are not the best way to tell.

As long as you will have the batteries out, get them Load Tested they may seem to have a charge but under load you may find that they have a bad cell.
I always Load Test Mine, even when buying new ones.