With the shore power back on, the batteries should recharge.
The description of the problem sounds like it could be an automatic transfer switch issue. Automatic transfer switches are suppose to default to the generator so that even with being plugged in to shore power, the generator should start and after it is running at full speed, the ATS will put the coach on generator power.
Whenever starting any generator, be sure any significant loads are turned off. Only turn on air conditioners or microwave, toaster or space heaters after the generator is running at full speed.