I've always been under the impression that a Honda 2000 would run a 15K AC if it had a Micro Air soft start capacitor. I would make sure everything else is off, including your invertor.
No, the 15K takes more constant current than the eu2000 is designed to deliver. I have the exact same scenario.
EDIT: 15K AC in my RV takes 14 amps continuously. The Honda eu2000i is rated at 13.3 amps continuous output. However, An eu2200i will supply 15A continuously so if that's what you have, you're good to go.
Not likely. IIRC, the EU2000 is only able to supply a little over 13A. That might run an AC, but it won’t start it. Especially since you need to run the converter too.
John & Kathy 2014 Grand Design Reflection 303RLS 2014 F250 SBCC 6.2L 3.73