The generator only puts out 20amps, not 30amps. And your Energy Management Center sheds loads in a specific sequence when there isn't enough power to go around. If memory serves, the A/C is the first load that is shed.
So my guess is that since the A/C draws 15 amps all by itself, you've got too many other things running that add up to more than 5 amps between them.
When you start up the generator, check the EMS panel in your overhead locker, and if it hasn't automatically switched to 20amp mode, then manually switch it.
Make sure your water heater is in propane mode, not electric, if it's on at all. And turn off everything else except the fridge when starting the A/C, then turn on one item at a time until you find out which load was the straw that broke Onan the Camel's back, i.e. the one that caused the A/C to quit.
Engineering: Finding complex solutions to simple problems you didn't even know you had.