We can't talk about your legal situation, sorry.
Strictly on the tax laws, the IRS will only award the exemption to the parent where the child lives more than half the year, no matter what has been agreed to in court. The IRS will not award the exemption to the non-custodial parent unless the custodial parents signs the release form 8232. Of course, the state court can order you to sign the form. You seem to be asking us for advice on whether or not you should sign the form, even though the court order is unclear? We can't give legal advice.
From a tax point of view, you still qualify for head of household even if you sign form 8232. For most tax years, you would be talking about the child tax credit which is a maximum of $2000. For 2021, you are talking about the child tax credit which has a maximum of $3000, plus possibly the recovery rebate of $1400, so there is more incentive for your ex to fight, and more for you to lose (if you fight and lose) or to give up (if you voluntarily sign the form).
The mother might fight for what is right. The mother might offer to settle for a cash payment that splits the difference. The father might want a declaration that the mother must sign the form every year, this might not be just about 2021. The court might view the case differently in 2022 if the father has given up custody since 2015.
Tough situation no matter what you do. We would always advise you have your own representation, even though you said you didn't want to pay for it.
