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Yes, only one of you can claim the child. Who will claim the child is a decision between the two of you. For MFS, if you are using online TurboTax Online to prepare your returns, you will need to prepare two separate returns and pay twice. If you use TurboTax Desktop allows you can prepare different scenarios and if you then decide to continue with MFS, then you can file both Federal returns without an addition Fed fee (however, you may be subject to additional fees for state filings).

If you choose to file married filing separately, both spouses have to file the same way—either you both itemize or you both use standard deduction. This may result in a higher overall tax tax that if you filed joint. Your tax rate could be higher than on a joint return. Some of the special rules and loss of tax deductions and credits for filing separately include: 

https://www.irs.gov/pub/irs-pdf/p17.pdf page 21

1. Your tax rate generally is higher than on a joint return.
2. Your exemption amount for figuring the alternative minimum tax is half that allowed on a joint return.
3. You cannot take the credit for child and dependent care expenses in most cases, and the amount you can exclude from income under an employer's dependent care assistance you are legally separated or living apart from your spouse, you may be able to file a separate return and still take the credit. For more information about these expenses, the credit, and the exclusion, see IRS Publication.
4. You cannot take the earned income credit.
5. You cannot take the exclusion or credit for adoption expenses in most cases.
6. You cannot take the education credits (the American opportunity credit and lifetime learning credit) or the deduction for student loan interest.
7. You cannot exclude any interest income from qualified U.S. savings bonds you used for higher education expenses.
8. If you lived with your spouse at any time during the tax year:
a. You cannot claim the credit for the elderly or the disabled, and
b. You must include in income a greater percentage (up to 85%) of any social security or equivalent railroad retirement benefits you received.
9. The following credits and deductions are reduced at income levels half those for a joint return:
a. The child tax credit,
b. The retirement savings contributions credit,
10. Your capital loss deduction limit is $1,500 (instead of $3,000 on a joint return).
11. If your spouse itemizes deductions, you cannot claim the standard deduction. If you can claim the standard deduction, your basic standard deduction is half the amount allowed on a joint return.

If you live in a community property state, you will be required to provide additional information regarding your spouse’s income and allocate community income. 

You can’t list any medical expenses from your spouses HSA on tax return as tax deductions as they were paid from his HSA. If he is claiming the child, he will list the medical expenses for the child. 

Taxpayers using the Married Filing Separately status do not qualify for the dependent care credit.

In addition, funds put into an FSA for dependent care are already "pre-tax".  This means you've already gotten a tax benefit for these amounts.  Therefore, no additional credit would be allowed for the same amount.  If you have expenses in excess of the FSA and use a different filing status, then you may still qualify for a dependent care credit.

In regards to putting money into both of your FSA's, the rules say taxpayers that are filing separately can put up to $2,500 into their FSA.  This means that each of you can put the maximum of $2,500 into your Dependent Care FSA; however, you can only use your dependent care expenses one time when requesting reimbursement.

Just because you utilized the FSA for your child, does not mean that you need to be the one to claim the child. However, when you prepare your tax returns, if you do not claim the child, you should still enter them as a dependent and indicate that the other parent will claim the child.  This allows the child to show up in the Dependent Care Credit section for determining if you had any excess FSA benefits that may be taxable.