It'll probably start and run. If it has a hard time starting you can add a $10 hard start capacitor to the AC. Having said all that, it might not be good for either your AC or other electronic item because of low voltage.
Having just gone through this with Yamaha generators I would suggest that you purchase the TT that you like and then size the generator to that. You can always add a second Honda which will run a 15,000 but AC.