He cannot file as head of household because the child does not live with him. You can file as head of household because you are the custodial parent who the child lives with. If you file as head of household it does not affect the other parent's ability to pay day care expenses.
You are the one who decides who claims the dependency exemption. You can claim it yourself or relinquish it to him. Regardless of whether he claims the dependency exemption or not, he cannot claim the child and dependent care credit on his tax return. It would be common in a divorce for the divorce decree to award the dependency exemption to each parent in alternate years. So it would not be unusual for a parent to pay child care expenses and have them deducted from the pay check in years when they don't claim the child on their tax return.
