He cannot use an FSA unless he claims the child as a dependent. If he uses the FSA to pay for care and does not claim that child or those children as dependents, then the FSA will be added back to his taxable income and he will lose the benefit of it.
If you are living together unmarried with two children, then most of the time, your best tax result will be for the parent with the higher income to claim at least one dependent. This allows the parent with the higher income to also claim head of household filing status, which has better rate tax rates than single. If the parent who pays more than half of the household expenses does not clean at least one dependent, then both parents must file as single and you lose the benefit of head of household status.
Whether it would be better for you each to claim one child, or for the higher earner with the FSA to claim both children, would have to be tested using different test tax returns.
