That 3000 watt genny should run a 13,500 BTU AC without a problem, as long as all other 110 volt items are turned off, as well as 12 volt items. With it being able to start the AC with the batteries disconnected, sounds like maybe the batteries are discharged, and a lot of the amps is going through the power center to charge the batteries. Next time charge the batteries some 1st, cut off all 110 and 12 volt items (and fridge on gas only like you are doing), start the AC fan on low and let that run for a minute, then start the AC. I take those precautions with my Yamaha EF2400Is, and it will run the AC without a problem. Surely, a Honda 3000 should do the same.
Chap , DW Joy, and Fur Baby Sango
2006 Ford F150 Super Cab 4x4
2008 Surveyor 263