Sounds like you may have the wrong type of inverter, but yes you can add an external transfer switch to bypass the inverter when on shore power. I did that with our old rig after I had an inverter go bad and keep me from using anything that was run thru it. I did it so if the inverter failed I could bypass it. In your case you could use it to bypass the inverter until you wanted to use it. The problem you are having we never had with our 4,000 watt inverter.
Dick, Joyce, Diego, Picatso and Gustav
2017 3720 RL, and 2013 HC 343RL
Pullrite Hitch, IS, Disk Brakes, 3rd AC, Winegard Traveler, Bathroom door mod, Dometic 320, couch for desk swap, replaced chairs, sun screens, added awnings, etc.