How to mitigate voltage drop?
Edit: 
I hadn't quite realized the pump draws 8 A at 110 V, which would mean 125 A at 12 V. The wire gauge is thick enough for that (4 gauge is 20 mm2) but not by a wide margin. At these power levels, minimizing the resistance is critical, all connections have to be perfect.   

To doublecheck the wiring: measure the voltage   


between the battery terminals  
on the input of the inverter  
on the output of the inverter  


all while the pump is running. A difference between 1 and 2 indicates a wiring problem. If 3 is much lower than the rated voltage, that will cause the inverter to shut down. 

Have you checked if the battery is properly charged? A voltage drop like that can indicate the battery is almost empty. When it's not providing a current, battery voltage will look okay, but voltage will drop rapidly when current is drawn.

If the pump really draws 880 W, that's a pretty big load (comparable to a starter motor), and on the edge of what a battery can supply. So that may be part of the problem.