As long as you both qualify, and you both agree to abide by your decision, the IRS tiebreaker rules don't matter. The tiebreaker rules only matter if you disagree about who should claim the child. Your incomes don't matter.
The only way to know which way gets the lowest tax and highest refund is to test both ways.
Someone in the household must pay more than half the household expenses. It only has to be $1 more than half, but someone must pay more than half. This is usually the parent with the higher income, but not always, if the person with the higher income is saving their money, or is spending it outside the household on other expenses (such as child support or alimony).
If the parent who pays more than half the expenses claims the child, that parent can also file as head of household, which has a slight tax reduction compared to single. The other parent files as single with no dependent. Or, the parent who pays less than half the household expenses can claim the child, but they have to file single, as does the other parent (because even though they pay more than half the expenses, they aren't claiming a dependent).
Those are your options, the only way to know for sure is to test it both ways.
For 2021 only, there is the added complication of the advance child tax credit and the stimulus. If parent A received the $1400 stimulus check, but parent B claims the child on their tax return, parent B willl get a $1400 rebate and parent A will not have to repay the stimulus. You are allowed to "double dip" in this case.
If parent A received the advance child tax credit (generally, $250 per month starting in July), but parent B claims the child on their tax return, parent B will get the full child tax credit. Parent A might have to repay the advance payments, but parent A will not have to repay the advance if they are low income and qualify for repayment protection. This is another case where double dipping might be allowed.
Turbotax will include all these factors. If you are filing online, you will want 4 separate online accounts with different passwords. Parent A plus child, Parent B alone, Parent A alone, and parent B plus child. Only file the two returns that match the situation you choose. AND, when preparing the returns, the return that does not claim the child should not enter the child's name in any form, and the parent who is claiming the child should answer "no" to any question about custody agreements, because that only applies to parents who live apart and share custody.
