Just to be clear, you cannot go directly into an inverter with shore power (on AC input side) without a built in transfer switch. At least not without causing a fire or frying something bad.
You can take the inverter, and I think this is what you are saying, hook up to batteries (with fuse) and then AC output as a cable/plug that can plug into your shore power on the outside of the trailer. This will work fine, and yes turn off the converter.
This is more complex than what you were mentioning but shows where and how a transfer switch is used. (sometimes this can be built into the inverter to simplify wiring)
This is one of the easiest to understand diagrams I have found.
https://i.imgur.com/9lcn6Lr.png
It is from this thread,
https://diysolarforum.com/threads/wh...s-right.20224/