Just remember your TT only has a 30A circuit. Usually the factory 30A cord is 25-30' long. If you are running the A/C, Microwave, Coffee pot, Toaster, and your distribution center is probably charging the house battery at the same time. The combination of using more than one of these devices at the same time can easily trip a 30A breaker either in the TT or on the generator. Adding an extension cord only adds to the load and makes it worse.
Your generator is more than capable of running your TT. The problem is you are maxing out the 30A circuit from time to time. Going to a larger generator will not help as you are still limited by the 30A circuit in the TT.
You need to buy or borrow a DC amp meter to measure the amp draw of your TT. My guess is you are maxing out at 30 amps or more causing the breakers to trip.
Also, if you take off the cover when you clean your A/C filter there should be a label inside that will say whether you have a 13.5 or 15K A/C unit.