Your ex can't claim any dependent care credit unless they claimed the child physically lived in their home more than half the nights of the year, regardless of who technically has custody. This will block you from claiming the child as a dependent. Even if your ex immediately files an amended return, it won't clear the e-file block to allow you to e-file. Print your return and file by mail as soon as you are ready.
Note that if the child lived in your home, your ex can't claim the child as a dependent at all unless you give your ex a signed form 8332. This allows them to claim the tax credit only, and they have to indicate in their tax program that the child does not live with them and they did receive a form 8332 from the other parent. It probably is not enough for your ex to remove the dependent care credit, they will have to delete the child entirely unless you give them a signed form 8332.
And if you did give the other parent a signed form 8332 to allow them to claim the child as a dependent, you must make sure you indicate that in your tax return, otherwise you may be getting an improper credit that will cause the IRS to launch an investigation.
