Originally Posted by LionFRR
Yes, the generator is powering the 2 A/C's.
That is a start but go back to the post from Cavalier. Where in the circuit is the battery charger connected? As he said (and this is NOT a good idea), some times the battery charger is connected before the Shorepower/Generator switch. Basically that means the charger is connected directly to the shore power line so when running the generator the power isn't reaching the charger.
Also as colemj mentioned, perhaps the power out from the generator is not clean enough for the charger's circuits and it isn't starting.
First step is to check for power at the input of the charger. Do you have a volt meter? Do you feel competent working around 120 V AC electricity? Be careful, this can be deadly on a boat if you aren't careful?
If you want to test this, check voltage at the input to the charger with shore power connected and then with the generator running and see if you show power with both.