Only the custodial parent can claim the child care credit. Your son's mother is the custodial parent, because he lived with her more than with you.
But she can't claim the credit because she didn't pay for the daycare. Since you're making unwritten agreements, why not make a deal to give the money for daycare to his mother and let her pay for the daycare? Maybe she will then agree to give you some (or all) of the money that she saves on her taxes.
