I second that. I would claim them. If he is not current...his problem. He won't mediate...his problem. He won't get CS adjusted....his problem. He would just try to bully you in other ways if you back down on this issue. If you have it in clear writing, and he is in violation of CS which gives you the right to claim the kids, do it.
I made a deal with my XAH this year to claim my daughter even though it was his year. It's to offset some of the CS I forgave. It did not come close to making up for that amount, but it helped some.
I want to come in and add, if you are afraid of some other thing in court b/c of this as an outcome, that would be a separate issue, and one to discuss with your attorney to see if it's worth it or not. I know your area has some messed up court rulings.