As others have said....20-amp will be enough to run the AC.....However, as you have stated, make sure water heater is on gas and maybe the fridge too. Microwave is out of the question and you might need to shut AC off if the Misses wants to use a hair dryer. You can further reduce your electrical load by replacing your interior lights with LED's to save some juice. Not a cheap upgrade, but if you do a few camp walmarts or limited on how much power you have on hand, every little bit helps.
On the generator issue....the new honda handi 3000 should be more than enough to fire the 13.5K AC. I will hopefully upgrade to one later this year as my 2000 series does not have the output to start my 15K AC unit. I did manage to borrow another 2000 series last year and run them in tandem and those two powered the AC just fine, but I also minimized my amp usage while I was dry-camped. Good luck....
2015 Rushmore Springfield
2011 F350 4x4 LB 6.7 Dark Blue Pearl