You are not the only one getting more confused by the minute....
You've written several times that you have a 50-amp shore cord, and a 50-amp extension cord. But now you are telling us that the 50-amp cord connects to your bus with an L14-30, which is a termination that is only rated for 30 amps. Between that and the generator-shore parallel connection with no switching, I have to say that your entire electrical system is suspect, and I don't think we can make any assumptions at all about it without a good, thorough inspection.
Probably not what you want to hear, but I would start by addressing some of these major deficiencies, and trying to diagram out just exactly what you have. With things as mis-wired as they are, there is no telling what sort of results you will get from any shore connection arrangement at all.
Right off the bat, I would (1) remove the shore and generator parallel connection, either by installing a transfer switch (manual or automatic), or by installing a 50-amp receptacle on the generator and using the shore cord itself to make the switch and (2) replace the L14-30 with a proper 50-amp connector. Then I would meter out all your cords and adapters to make certain they are properly wired. Once that's done, you will have a good starting point to diagnose the A/C problem.
All that said, I am sticking with my theory that the shore cord is not making good contact with the dogbone, but the extension cord makes good contact at both ends. Your meter may show good voltage, but that is at a no-load state -- adding a load can cause significant voltage drop over a marginal connection, enough so that the A/C will not start. (Running through the under-sized connector is not helping this situation any.)