The every other year thing requires zero cooperation if (IF) the IRS Form 8332 are signed in advance. Or if you get every year.... have her sign the forms in advance giving you the right to every year.
1) She isn't going to do that voluntarily.
2) A judge isn't going to force her to do that if the condition of the agreement is that the person making the most money gets the exemption.
3) I would advise that anyone fight that particular "order" tooth and nail.
There is a classic bit of federal tax case law out there that explains WHY you don't do that.
Dad was ordered to pay child support. Mom had primary custody. The judge ordered that dad got the tax exemption and mom signed a form 8332 for all future years.
Dad immediately quit his job and disappeared. Mom eventually went back to court and got sole custody, and the judge ordered that she got the tax exemption. Every method was adopted to try to collect CS from dad, but no one was ever successful.
Dad however, kept claiming the child and the IRS (and later a federal tax court) ruled in his favor. Why? Because mom signed the danged for 8332 for all future years. The federal judge stated in open court that it was grossly unfair, but that he had no choice but to rule in dad's favor, because there is no methodology to rescind a form 8332.
So, I will tell anyone who is contemplating signing it for more than one single year at a time, NOT to do that.