I have a similar issue with my NTU 2005 Starcraft. The AC has a dedicated 20 amp outlet under one of the seats and right behind the converter box. But when I'm plugged into the house current (15 amp), that outlet is dead and the AC won't fire up. I bought a 20-15 amp adapter pigtail just to make sure it wasn't the AC unit, and sure enough, it works fine when I plug it directly into the house current (although I didn't turn it up high or for very long).
Although I haven't run the full set of wiring yet, the breaker and outlet both seem fine, and the other 110V outlets work fine. Thought it might be a GFI issue, but that's not the case either.
Is it possible that the converter is set up to not make the 20 amp outlet 'hot' unless there's enough amperage to safely run the A/C unit? Such as when plugging into 30-amp at a campground? That's our next step, but frankly we mainly boondock so the A/C isn't something we'd use often...but nice to have.