I would say, "ok, fine, I don't use those products anyway". How are they going to know what products I take at home, lol. As long as I get my religious exemption.
No, do not go down that path.
Once you sign on the dotted line you have signed a form that can be used against you in a HR action.
Think of this as another way that the authorities inflict papercuts by the thousands.
They are not looking for honesty but rather a way to gain a new power over one. I'll recommend that you review the federal rules on religious exemptions and the battles previously fought.