Dave,
I had/have the what sounds like the same problem. What it boiled down to is the Progressive Dynamics transfer switch. The switch monitors voltage on one leg of the transfer switch (on the generator side). When the main a/c is turned on, the the generator voltage drops below the switches acceptance level (around 100 volts). The voltage drops because as the air conditioner compressor tries to power up, it draws a lot of power (around 50 amps for less than 1 second). Because the genset is a limited supply of power, the voltage drops for that same instant. During that duration, the transfer switch drops out generator power. If you are on line power, it will switch to that or if no line power is available, it switches to a neutral state. After several seconds, the transfer switch see's generator power again at an acceptable level and tries to switch back to generator power. If the air con is still switched on , the compressor tries to turn, amps surge, voltage drops and the transfer switch drops out again.
I bet you here several loud thumps from the underbelly as this is happening.
If this sounds like the same problem you are having, PM me and I can tell you what I have done to fix this.
John