I haven't been around the 9000BTU roof units but my 2KW Honda runs my 8000BTU house Window unit just fine.
Sometimes our kids like to follow us to the local beach camping and they setup their tents. They bring along a house window unit 5000BTU air conditioner and setup on shipping crates just outside their tent and they run just fine on the 2KW Generator as well...
I don't think my generator would mind the 8AMP draw. I suspect however there will be more amps on the initial startup to overcome. On my roof mounted 13,500 unit if I turn off everything around the trailer I have got my air conditioner to start up. It does trip the breaker on me everytime however when it tries to recycle hehe..
Roy Ken