Why Won't My 1000W Inverter Run my 800W Coffee Maker?

Full Mosey hit it right on the chin...

Either there are enough volts at the inverter terminals or there isn't.

This is 99.9% of the fundamentals of what is lacking.

AFTER that has been determined, then in a separate check, do the battery terminal voltage.

CCA tells you what is available at 9.6 volts. At 9.6 volts any inverter will stick it's tongue out and tell you to go screw yourself.

AMPS DELIVERED AT 12.0 volts is a more rational test.