I think I may have found my answers, now that I understand it better. The 6500/5500 has (2) 20 amp and (1) 25 amp outlets. The 4370/3500 has 15 amp outlets. My current generator has 13 amp outlets.
According to google, a 13,500 watt AC takes about 1250 running watts (or 10.4 amps). This gives me 2.6 amp (or about 300 watts) at max. This explains why it is "popping". It is reaching 90% of its max often and simply breaking the circuit.
Based on the above, I should be OK running the AC with the 15amp per outlet generator, but I won't have much if any left over, so for $89 more, the 20/25 amp per outlet generator will give me 2400-3000 running watts (on just one side). Even if I get 85% of this in the real world, this is as much as I need at anytime and they sell them on sale for $400 (or less).
So I have to decide how much money I want to spend and what I want to sacrafice (noise).