There is no such thing as "50/50 custody" in tax law. Under the tax law, the parent that the child lived with for more nights during the year is the custodial parent. Your divorce decree or other agreements cannot override the tax law. The custodial parent is the one who can file as head of household. So if you don't know which of you had the child for more nights, figure it out the best you can, as long as you and your ex both agree on which of you it is. And starting January 1, keep track of the nights for 2025. Mark it on a calendar.
The other things that you mentioned - claiming the child as a dependent, paying child support, and AGI - have nothing to do with who can file as head of household.
