Do all of you live together as a family? If so, then the child should only be entered on one tax return, not both. If the other parent already filed and claimed the child then you cannot e-file a return with the child's SSN on your return. You need to delete the child's SSN and just file Single with no mention of the child on your return.
That does not change anything about how you have to file your tax returns. The childcare credit cannot be entered on a second tax return. Again---if you all live together then the child can only be claimed on one tax return. So the two of you need to work out between yourselves how to share the refund that will come in only one name.