It certainly is possible (and not uncommon) to plug the camper's shore power connection into an inverter, or to do something equivalent electrically, and power the whole camper from the inverter. It's easy and inexpensive to install that way.
You do need to be careful what is run from the inverter, and remember to turn off potentially a fair few things before starting up the inverter. You know about the converter. You'd also want to make sure the air conditioner is off, and the water heater electric element (if you have one), and the fridge is set to gas only mode rather than auto mode, and any space heater or fireplace turned off, etc.
A more foolproof approach is to power only certain circuits from the inverter, using a transfer switch and a subpanel (one or both of which may be built into the inverter).
The microwave, toaster oven, and coffee maker all consume a lot of power, and so require a pretty hefty battery bank for more than a nominal amount of use. The latter two at least are probably better replaced with the gas stove and oven (and appropriate cookware) when boondocking.