I suspect Amnesia is right. But check your connections and make sure there is no high resistance anywhere. Get everything clean and tight. It might make a difference. Otherwise, more or bigger batteries.
Assuming a perfectly efficient inverter for simplicity, and zero line losses, 1000 watts at 110VAC would be... 9.09 amps. The DC side, at 12VDC, would be oh... 83 amps. Quite a bit of current
. When you DO figure some line losses and internal resistance in the batts, you could have a significant voltage drop which might be enough to trip most modern inverters. Don't pick at my figures I know they are imprecise. Just showing how you could have a high enough current on the 12V side that a very small resistance could be enough to make a significant voltage drop. I might not be illustrating this quite right but I think you get the idea.