Thank you everyone for your words of understanding and encouragement.
Our current order stipulates that he pay around $400 per month; he just never pays it, and because he is completely unemployed and living with his mother for free, I'm sure they will lower this amount considerably. I spoke with child support today, and his failure to pay has no bearing on the review.
I had never considered asking him to sign away his parental rights in exchange for no child support. It would be interesting to see if his ego will override his common sense. I imagine this would require me to hire a lawyer. Does anyone else have experience with this?