If you had custody more actual days than him,, and you can prove it, there is no reason for you to file an amended return and anything he or his CPA wants you to do may violate the law. You may need to get your attorney (if you have one, and if they actually understand these tax issues which some don't) to get involved.
If you had custody more days than him, you are the only parent entitled to claim the child as a dependent.
You can sign a release form that allows the other parent to claim the dependent exemption and child tax credit. This will cost you a variable amount of money depending on your overall tax situation. Some custody orders require the custodial parent to release the dependent exemption, for example every other year. The IRS does not get involved in enforcing custody orders. If you have such an order in place and you don't sign the release form, the IRS will not take action. The other parent might be able to get the family court to order you to sign under threat of contempt, but the IRS won't enforce the order, and the non-custodial parent can claim nothing without the release.
If you don't have a custody order or the order is silent on the subject of the dependents, then there is no reason to sign the release unless you are feeling generous. If you did sign a release for 2016, you would have to go back and amend your return to reflect that fact.
But, even if you sign the release, it will only entitle the other parent to the child tax credit and the dependent exemption. The ability to use a tax-free FSA for child care expenses always stays with the custodial parent and can't be waived, released or transferred. (Likewise, the ability to qualify for head of household status and EIC also always stays with the custodial parent and can't be transferred, waived or released.)
The only way he could claim the tax-free benefit of the FSA is if you completely removed the child from your tax return. (It should cost you a lot more than $600 unless you are in a fairly unusual situation.) Then he could file as if he was the custodial parent. I don't think you would be breaking any laws but he and his CPA would be.
Unless the CPA asked you to sign a release form, they are probably trying to file falsely. (Does he really have a CPA who is advising him to file a false tax return, or just he just say he has a CPA to try and convince you. Have you ever seen this person?)
If the CPA did ask you to sign a Form 8332 release, then it may be the case that they are planning to file honestly and he will save more than you lose and is willing to make it up to you. But that won't cover the FSA. It would give him the exemption and the child tax credit only.
