No.  The IRS will not honor your court order.  (And, while they will honor some court orders issued before 2009, they never honor a court order that contains a contingency.)

The only parent entitled to claim a child as a dependent is the parent where the child lives more than half the year.  The non-custodial parent can only claim the child if the custodial parent signs a release form.

You may have a case to sue your Ex in the family court to enforce the order, but the IRS will not enforce it for you.

(As a practical matter, you are better off negotiating a $2000 reduction in child support every other year, which is the approximate value of the dependent exemption and child tax credit, rather than having to fight over the exemption every other year.  More attorneys need to know the tax situation better.)