Alright, so an update for everyone:
-Battery was fully charged earlier (I even tried it with the converter CB off)
-The only loads going at the time were the 12V ceiling fans
-I'm at sea level and the temp is in the high 70s
-Generator ECO mode was OFF
-Water heater was OFF
-Generator is brand new
-I did not try running the fan first then starting compressor (I may do that next)
-I did confirm that there isn't a start capacitor (I already ordered a SUPCO SPP6 from Amazon--for $10 it is a cheap possible fix)
Was there a running joke about start capacitors that I am missing? ("Worst answer yet.")
Also, the dealer said that "a 3500W generator is required to run a 13500 BTU air conditioner." That is counter to what I see most other people saying on all th forums, right? (I see people saying they get their A/C running on a 2000W under good conditions.)
Hopefully this isn't a bad generator!