On mine, running is between 10 A and 14 A. Inrush is hard to measure, but was above 50 A. Now it's more like 30 A. But it really depends on how well that start went. My roof top AC has many many hours on it, likely 5k+, and I could probably stand to replace the compressor (in other words the whole thing).
A 1kw generator will be a challenge. I'm working with a 2kw unit, but also a 13.5k btu ac (which will chill an eb van full of gear, and is more than one might spec with just humans in it).
If I was in your situation, I'd honestly just step up to a 2k generator, since those really aren't that different in cost, weight or noise to 1k units. This seems to be the same in both name brand (cough, honda eu) or generic.
'99 EB ex ENG KSWB news van, low rent 4x4 conversion (mostly fixed by now), home built interior.