Some small details may change the answer?
Do other 110AC items also fail when the air fails running on generator?
My first thought is that the generator power and the cord power are much the same except generator has a lower limit. The limit should not be a problem but there are fairly common problems with the connections in the tranfer switch coming loose and the contact is not always good.
Click this drawing to see what shows in the transfer switch:
Basic idea is that the switch chooses cord or generator power and then sends that out to the load center. Once that choice is made, the rest is the same used for either.
So if the power fails to the air cond. on generator, it should also be bad to all other 110 AC items like microwave, outlet, etc. If those are also bad, I would suggest checking the connections inside the transfer switch for them getting somewhat loose to make or lose contact at times!
Some suggest checking and tighten these connections as a good preventive measure! The three I highlighted in blue are the three main ones to check but while there give the others a check as they may also be getting a bit loose.
One of the problems is that the soft copper of the wire can get mashed and change shape as we drive and bang them around on bumps!
Hope that may help?