You are simply not allowed to do what you are trying to do, because you did not live apart for the whole 2nd half of the year. That is, you may not split the tax benefits. Only one of you can claim the child on his/her tax return for 2022. If you can't agree on who that will be, then the rules say the parent who had the child the most time gets to do it. Even for next year, you can not split the benefits, in the manner you proposed ("I am claiming my son as HOH and dependent and she is claiming the child with the daycare expense").
There is a special rule in the case of divorced & separated (including never married) parents. When the non-custodial parent is claiming the child as a dependent/exemption/child tax credit; the custodial parent is still allowed to claim the same child for Earned Income Credit, Head of Household filing status, and day care credit. This "splitting of the child" is not available to parents who lived together at any time during the last 6 months of the year; then only one of you can claim the child for any tax reasons. The tax benefits may not be split in any other manner.
Note in particular that the non-custodial parent can never claim the Earned Income Credit, Head of Household filing status or the day care credit, based on that child, even when the custodial parent has released the dependency to him.
Ref: https://www.irs.gov/publications/p17#en_US_2017_publink1000170897
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For the IRS rules, Custodial parent is the one who has physical custody, not legal custody, and there is no such thing as joint custody.
