3kw generator is going to be very marginal for a 15k btu AC in combination with other general loads in the TT as you know from your previous experience of it struggling. You've had your TT a couple weeks now and believe it or not just having the converter running hard to recharge your battery can be close to 1kw making it impossible for 3kw to start the AC.
Turn everything off with a 110 AC breaker and the converter (probably have to get behind a panel to unplug it) then try the AC Unit. If it works you are going to have get a hard start capacator for the AC (might be just enough to work) or get a larger generator.