Basically, and with no intention of sounding rude or anything, the IRS could care less about your custody agreement. It means nothing to them. The IRS has their own rules you must follow and your custody agreement can't change that either. Only a federal judge can change or alter the IRS rules. But since federal judges don't deal with custody agreements, that's just never going to happen. Here's the IRS rules for divorced, separated or unmarried parents.
The IRS defines two types of parent. There's the custodial parent, and the non-custodial parent. The custodial parent is defined as the one with whom the child lived with for "more" than 182 nights of the tax year. Those nights don't have to be consecutive either. Furthermore, the custodial parent also provided more than 50% of the child's support for the entire tax year.
The non-custodial parent is defined as the one who does not meet the requirements to be the custodial parent.
Therefore, it is the custodial parent who has the legal right to claim the child on their tax return. But for every rule there is an exception of course.
If the custodial parent is allowing the non-custodial parent to claim the child, then the custodial parent must provide a signed IRS Form 8332 relinquishing their right to claim the child, to the non-custodial parent. This area is where a lower court judge "can" get involved. If the custodial parent refuses to sign the form 8332, then the lower court judge can find the custodial parent in contempt of court and either fine the custodial parent, incarcerate the custodial parent, or both until the custodial parent signs the 8332. But the button line is, regardless of what occurs the non-custodial aprent flat out can not claim the child without a signed 8332 from the custodial parent. Under no circumstances will the IRS get involved in any legal matters concerning this either.
So, if you are the custodial parent and it's your year to claim them, you claim them. If your e-filed return is rejected by the IRS because they were claimed already, then you have no choice but to print, sign and mail your return to the IRS. Then when the IRS sees the dependent(s) are claimed on two returns, both of you will be audited and the loser will pay the back taxes, fines and penalties.
