The FSA (Flexible Spending Account) is for healthcare cost. Per IRS Publication 502 page 15 second column:
You cannot include in medical expenses amounts you pay for the care of children, even if the expenses enable you, your spouse, or your dependent to get medical or dental treatment. Also, any expense allowed as a childcare credit cannot be treated as an expense paid for medical care. http://www.slu.edu/Documents/hr/benefits/IRS%20publication%20502.pdf
So I 'think' you may not be understanding things the way you need to. You really need to see what your specific FSA plan is for. But overall if your plan allows it, I see you already understand you can't double dip here.
You may also want to read tax topic 602 at https://www.irs.gov/taxtopics/tc602 to understand the limits and how your income can affect your ability to claim "any" child care credit regardless of how it's paid.
