Wire inverter to existing transfer switch or need another?

When dry camping, could I run a power cord from the inverter to the shoreline cable with an adapter thus using the existing transfer switch. Limiting it to 15 amps with a resettable power strip. Only problem may be that the fridge will be on the inverter's 110v instead of propane. Then when I turn on the generator, the transfer switch in the MH should isolate the inverter. Right???? Also, could my DW use her hair dryer at half speed and heat and I can use the microwave at half power , obviously not at the same time.