No, you are not allowed to claim both children on the Child Dependent Care Expense Form 2441.
See Can I claim half of my child daycare even if the other ... for the detail reasons.
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Yes, you only claim the credit for the child you are claiming. If you are not claiming the other child as a dependent, you may not claim any other tax benefit, like the day care credit, on that child.
If you and the other parent live together, either one of you (but not both) may claim the child. You may decide between you which one will claim the child. Only if you can’t agree, do the IRS tie breaker rules apply, to see who has first choice. It may be worthwhile to prepare trial returns, both ways (three ways in your case), to see which way the family comes out best. This tool may be useful: https://turbotax.intuit.com/tax-tools/calculators/taxcaster/?s=1.