I do the shore power cord thing and while it great that it powers the entire trailer, it power the entire trailer... Meaning yes you have to be aware of any loads on the AC system prior to hooking it up. I have blown a couple of fuses on my inverter as a result, but fuses are cheap and easy to swap out. Have to disable converter and put fridge on gas only mode.
I was powering our TV off my inverter last trip, and my son fired the onboard genny to run the microwave. It didn't seem to affect anything that I could tell