Another inverter/solar grounding question for the experts

I'm betting that your ground wire for the DC system is not up to the task. Using an 8 AWG ground,with 4 batteries and an inverter trying to pull +1500 watts, the inverter is seeing a low voltage condition. Your use of an extension cord may be just enough to allow some capacitance to overcome the inverter's bad feelings about the situation.
The frame ground from the inverter is just as MrWizard has stated, a protection and interference device, not part of the power profile.