To the OP:
We can't help further without Make/Model/Spec on the Generator.
About battery charging:
Some Generators charge the House Battery at a very low rate when running.
On Shore Tie, or with Generator running, your coach's Converter/Charger should charge the Coach (House) Battery.
With Engine Running, the Engine/Chassis Alternator should charge its battery (the Engine/Chassis one).
With Engine Running, the Engine/Chassis Alternator should ALSO charge the Coach/House battery.
This Engine Running/House Battery Charging is made possible by a small continuous duty Relay (possibly under the Hood, possibly near a Battery). It does this because a wire from the Engine's Ignition Switch energizes it to connect the House battery for charging while you drive.
This relay is also your Emergency Start. It energizes to connect Coach to Chassis batteries when you push that dashboard button. You're giving yourself a Jump Start without having to use Jumper Cables.