Signal trouble when on generator power

It sounds like the generator voltage is not stable enough. Have you tried running the D12 from a UPS, that should produce a stable power source..?

I've had the voltage checked.  They tell me it's stable.

There's more to "stability" than just checking for 120VAC.  The biggest problem with generators is AC line noise and poor frequency regulation.  A UPS can help isolate those issues.  Or a better quality generator. 

I ran a 16KW Westerbeke (diesel power) for years and although I didn't have Directv, both of the Panasonic TVs worked perfectly when fed a cable signal.  What specific issue are you having?  Does the issue stop when you're on shore power?  Do you have the coax grounded to your ground plate?  You might run the Gen with the unit on shore power.  That may tell you if the interference is RF or carried on the AC line.