This probably won't be a popular response with you, but devil's advocate and all that...

Can your ex afford to take care of your kids with less than the $1000/month from you for a year? If she doesn't work, that's not enough to live on with two children. Even if she does work, if your kids are young, that probably doesn't cover daycare, so she's got rent for a larger place, food, clothes, diapers, and all that other kid expense. Maybe she makes plenty of money and your contribution is just bonus. With my wife's career, that would be the case, so if that is your situation - fight this!

If it isn't, how much can you realistically afford to contribute? Regardless of how awful of a spouse she was (and she sounds like a real peach), she is still taking care of your children. I know I personally wouldn't want my kids to go without, and if that meant I had to sweep floors or flip burgers, I would. You may need to get a second job.

Good luck.