OK, first things to check:
1) Are the connections to the inverter good?
2) What is the voltage at the inverter input terminals when it shuts down? Ideally, measured right before it shuts down and right after.
3) Are there any settings on the inverter for low battery cut out?
Question 2 really is a check on question 1 - if the voltage at the inverter were, say, 12.6V right before it cut out, and then jumped up to 14V thereafter, I would suspect that either the wiring is too small or that there is a bad connection adding resistance.
Also, does the inverter have a low-load cut out? Will the inverter run with nothing plugged into it? Maybe the inverter is sensing that there is no load and shutting off?