Probably different Circuit
I agree with the other people that the outlet in question is probably not on the circuit that you re-routed through your inverter. While on shore power, plug a light into that outlet, then turn off each circuit until it turns off. I know that my hot water heater is under my sink, right behind my table, and they may have been lazy and tied into that line instead of the correct one.
Regardless, you should be able to turn off each circuit one at a time at the box and verify which breaker it is on. Then with that, you can come back and let us know what you find.
I did something similar, I installed an AIMS 2000w inverter, but I went ahead and just routed everything except for my AC through it. It was funny when driving down the road and the kids were watching a movie (maybe 30 minutes after we left the campground), then everything went off. Turned out, I had left on the electric water heater and the batteries and car alternator just could not keep up.
Hope you find your solution.
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