Leaping thru lots of assumptions. Here goes.
Your inverter powers the house circuits...all of them or a significant portion of them. In other words, the inverter feeds (let's say) the main panel, which in turn, feeds a significant number of the 120 volt outlets in the coach when the inverter is on.
If so, it MUST have a transfer switch to protect not only the inverter but the coach circuitry and appliances from what amounts to out of phase power coming from two sources in parallel. (
the power grid must be very carefully managed to keep all power in the alternating current IN PHASE."
Remember that shore power and inverter power are both alternating current. If they are out of phase, this can be very destructive if both are on at once.
And your inverter will attempt, very feebly, to back feed the power grid.
This is no different than a situation when one runs a backup generator thru one's main circuit panel without a transfer switch. One can back feed the grid with the generator and kill a power line worker who expects the lines to be dead.
Now the possibility is that your inverter has a built in transfer switch. When it senses shore power, it isolates itself and does not feed power to the coach's circuit panel. Some do and some don't. At the 1000 watt level, "it depends" is the key phrase. Check.
ALTERNATIVELY:
A smaller inverter such as yours may only run a couple of dedicated circuits that are NEVER powered by shore power. I have a small inverter in my PUP that is a truly stand-alone power system that has NOTHING to do with the shore power setup. I must move the appliance plugs to the inverter output to use that as a power source. If you are setup this way, then you have nothing to worry about.
More than likely, however, you are in the first scenario.
In that case, you MUST have a transfer switch, or your setup is an accident waiting to happen. Y
ou can get an inverter with a transfer switch built in for dirt. That's an illustration, not a recommendation.
If this is a factory-installed inverter, most likely it has a built in transfer switch. If it's an aftermarket, dealer installed inverter, your guess is as good as mine. The specifications with your owner's manual packet should include a pamphlet for the inverter. You also know the brand and model, so Google is your friend.
But the fundamentals of why you need a transfer switch IF YOUR INVERTER FEEDS EVEN A PORTION OF THE COACH'S MAIN POWER PANEL are important to know. When you're plugged into shore power, that power source is fighting with the inverter unless the transfer switch segregates one from the other.