Remove the shore power for both inverter and genny operation. The genny should be protected by an automatic transfer switch, but it is POSSIBLE for that to fail and do damage. Good idea to run a "Load" equal to 80% of the maximum inverter capacity for a few minutes to test it properly. Example for a 2000 watt inverter, run a toaster oven or hair dryer rated at 1600 watts. If you have a meter you can monitor battery voltage and line (120 volts) voltage durin the test.
2008 Itasca Meridian 37H & 2015 Flagstaff T12RBST
2011 & 2012 Len & Pat's "One lap of America"
27K miles & 41 states in 13 months - Woo Woo
Yellowstone Lake 6-1-2012