So I finally rented a Honda 2000 and tried running my Cool Cat.
Long story short, it works, at least in my driveway and at the current conditions.
I tested it in Iowa at around 900ft, with the outside temperature at about 80. With eco mode off, it started up just fine, after a very brief bit of strain on the generator. It just sped up for a second, then returned to normal once the cold air started blowing.
It will not run in eco mode.
The 2000 is louder than I anticipated, but I had it sitting just inside my open garage, so out in the open it might not seem as loud. In eco mode it's very quiet, but won't run the AC.
I did stop and restart the AC just to make sure it would, and it did just fine.
If you want to test it for yourself, Home Depot will rent one of these for a few hours for $40 or so.
Here's a short video of it in action: