The idea in the Mootorhomes is the same with RV Trailers to never let your batteries get down below their 50% charge state which is around 12.0VDC. If you have enough battery capacity to not let that happen then by all means leave your INVERTER on.
When I am running on my batteries I keep a good eye on the DC VOLTAGE PANEL and when it get to 12.0VDC then I start shutting things down. We RV trailer guys usually have to wait until 8AM the next morning before we can use our generators to re-charge our batteries...
If you are continuing to drop your batteries below the 50% charge state then I would power down things. Your batteries will not perform well for you if you constantly let them get below the 50% charge state and not get re-charged right away. You will eventually have to replace them or suffer big time poor performance.
I suspect your battery bank is a large one so do the tricks of the trade to take care of them. Worse nightmare for us is when it goes dark on us at 10:30 at night haha...
It would be neat to have an alarm go off when the battery DC voltage drops to around 12.0VDC like the BOGART Trimetric TM-2025-RV monitor does.
Roy Ken