First, must say that your case is the opposite of just about all others. Usually, A/C will start on 30A shore tie and it's the owner's generator that won't start it.
It seems to me that your household power outlet is deficient somewhere. You could have a 30A receptacle connected to a 30A branch circuit in a fully good 200A power center in the house. But if the connecting WIRE isn't rated for 30A (too small a wire gauge), your A/C will not be able to draw the power it needs.
Likewise, if your outlet is OK but you extend the RV's shore tie cable with a light duty extension cord, the A/C's starting demand won't be met.
Look at it this way: Your garden hose and the city water main out in the street might measure 75-PSI water pressure. That does NOT mean your hose can deliver anywhere near the gallonage that the main can. You can have voltage, but that doesn't mean you're going to get the amperage you need.
Do you have an AC (alternating current) voltmeter that you can set up in the coach and see what happens to voltage when you try to start A/C on shore tie? My guess is it's going to take a nosedive.
