Short answer to your question, no, I do not have an inverter back-up for my sump pump.

My neighbor had a battery back-up for his sump pump; it automatically cut in when the power went out. He was never satisfied with it because of the short life of the 12 volt deep discharge battery he had; at best in a real rainstorm (when needed most) he got 1 to 1 and a half hours life. I don't know the power draw of his pump. Our power outages typically run longer than that. On one occasion I had to run a long extension cord from my house to his to keep his pump running. He ended up with an automatic stand-by generator.

So, in addition to resolving the inverter issue, if you haven't already done so, calculate the power drain and how long the battery will last. Can you make it home or can someone else start the generator in time before the battery dies? You are probably already on top of this issue, but just though it would be worth mentioning.

I have run my sump pump on two different portable generators and the new auto stand by generator. Don't know their sine wave properties, but the pump ran fine. My impression is that motors are less sensitive to sine wave concerns than other electronics, but that is far out of my expertise.

Good luck.