Most genset setups have the generator running for some period of time - usually 10-15 secs - before the field kicks in and it starts generating electricity. Then there's a 5 sec delay before the transfer switch will switch over IF it's designed to favor generator over shore power. The only reason yours would not behave this way is if someone got into the wiring and screwed something up.
Just running the generator is ok, but you should be running it under load occasionally to dry out the stator windings. To do this you'll need to start the gen and switch off the shore power, then start something in the coach that will draw a few amps. I use the clothes dryer or a ceramic heater.
Flipping on the inverter before doing all this will prevent power outages that would require you to reset clocks, etc. Remember the 5 secs delay before the transfer switch switches to generator.