the use of a 20amp outlet should have no effect on your situation, as 120v power is 120v power, regardless of the 'size' of the outlet....and any 120v power should certainly provide charging to your house batteries via the Converter, or an Inverter with an integrated Battery Charger, depending on which you have.
As for the Alternator, motorhomes do typically have a way for the alternator to provide passive charging to the house batteries, WHEN the chassis battery(s) have reached a certain charge level - the 'extra' goes then to the house batteries.
For the generator - it 'starts' by using the house batteries for starting power, which you've seen. The generator may have a condition that is not allowing it then to continue to run, such as either low fuel level in the coach's tank(must be above 1/4), or a clogged generator fuel filter/line, etc.
As the generator runs, and after 20-30 seconds, the ATS(auto transfer switch) should then sense it's 120v output and 'click' or 'thump' into place, allowing that power to then get to your Main Breaker Panel, for power to with the coach.
When the generator stops, you may also hear a 'thump' as that ATS 'releases' it's magnetic transfer switch, sometimes even minutes later.
For your Batteries you 'installed', you didn't mention whether that's two 12v type or whether it's two 6v type. That's a big difference in how you then 'wire' them.
...2014 Ameri lite 259BH...
towed by our '14 Thor Palazzo 33.3 diesel pusher...