I am Jewish and H was raised Catholic. When we were getting married, I told him I would be ok with having a priest present along with the rabbi. It wasn’t going to influence my religious beliefs and I felt it would be respectful to my then future MIL. H went to visit the priest at his parish, who remembered him from when he participated in folk Mass. H told the priest of our plans and the priest said it would be okay as long as I agreed to raise the children as Catholics. H told him that we already had two children and our son had a ritual bris and our D a synagogue naming. At that point, the priest looked at him and said: “My son, you are far too immature for the sacrament of marriage.” H said “Well, given that I am 35 years old, I think that I will not wait around for you to decide I am grown up enough to get married.” We wound up having a cantor and no priest at our wedding, which was held in a hall, not a shul.
My cousin actually did sign the paper and had her son baptized. When he was about 5 years old, she discovered her H was cheating, threw him out and divorced him. She then burned his baptismal papers and had him bar mitzvahed when he turned 13. The ex remarried in the Church but I am not sure if he got to do that because my cousin destroyed the baptismal papers or because the marriage was declared invalid or if he just paid someone off. This was about 30 years ago. My cousin’s second H was an Orthodox Jew!
I have a friend who paid thousands of dollars to secure an annulment for her intended H. They were divorced within 3 years because he was a cheater, which was why his first marriage ended (well before my friend met him). I think part of the money went to a payoff to the first wife to say she never wanted kids.