You are correct that the person that makes the most generally gets the most benefit from the child unless it is an EIC issue.
You can use HSA and FSA funds for medical care for yourself, a spouse, or a dependent.
If you will claim the child as a dependent, you can use the pre-tax money in the FSA or HSA to pay for expenses. If you don't claim the child as a dependent, you can't use the tax-free money to pay expenses or if you do, the expenses won't be "qualified" and you will owe income tax on the use of the money.
According to the rules from the IRS:
If you are living unmarried with the other parent of the child, only one parent may claim the child as a dependent on their 2017 tax return. That is the parent who can also pay for medical expenses from an HSA or FSA or take the medical expense tax deduction or child care credit for expenses paid with after-tax dollars.
Assuming you and your partner will live together with the child, then either one of you can claim the child as a dependent, but not both -- a child may only be claimed as a dependent once.
