If you did not request reimbursement from the FSA custodian and the full $5000 is still in your account:
If I worked through form 2441 correctly, then you can indeed leave the money to be carried over to use during the 2022 grace period, and you can apply your care costs toward the credit, up to $8000 of costs for one child and $16,000 of costs for 2 children. However, make sure you don't also have an FSA for 2022 (because you would exceed the allowable amount) and make sure you can use up that $5000 within the grace period.
If you already requested reimbursement from your expenses:
There is no legal way to put the money back in the FSA. If your plan allows this, they are breaking the law. You would apply the first $5000 of costs toward the FSA, and could still claim a further $3000 (one child) or $11,000 (two or more children) toward the credit.
