Dig until you discover what fundamental "truth" the bigot rejects. That will usually be the opposite of some fundamental "truth" you fully accept.
Accept that people are rational. Reason proceeds from axioms, these are the "givens", statements of fact we find self-evidently true, meaning we do not think they require any proof, they are just true! We refuse to argue in circles about why they are true, we just cannot believe they aren't true.
Unfortunately, we don't all believe in the same set of axioms! Sure, we can all accept 1+1=2, but how about the statement that "God exists", or "An eternal soul exists and there is a life after death"?
I can get into arguments with my own family about those; and they usually end in "I just cannot believe that," by one side or another. I take those arguments literally, to mean the person I am arguing with has so much invested in believing their axiom, that it would be emotionally devastating and life-changing to reject it. e.g. My sister believes in the Christian God and her life is organized around that and she will die believing that, no amount of reason can change that, because for her the fact that God exists means all arguments to the contrary are flawed, and must be wrong because they violate a given.
Something similar will be for your anti-gay character. I personally am liberal, any kind of sex between consenting adults doesn't bother me; as long as nobody is manipulated, deceived or feels coerced. But in the end I am a "bigot" too, that opinion of mine is based on my own fundamental belief system, things I think are absolutely true. My opinion on this cannot be changed: I just cannot believe the fundamental changes that it would take to make me oppose homosexual sex, or displays of affection, or marriage or raising children or anything else related to it.
That is what you need to find for your character, and often it will just be the opposite of one of your own fundamental beliefs. You need to find what they cannot let go of, for whatever reason, that leads them to reason that homosexuality is wrong and must be prevented and is damaging society in some way.
That may be something personal to them: They can't stand the idea that their offspring will not reproduce and give them grand-children.
Or it may be a different kind of personal thing: God said so, and they can't stand the idea of the slippery slope that results if they start to believe the Bible can be wrong. It would make them question their faith, and their faith is a major component in their emotional stability, they've invested thousands of hours of emotional engagement with it. They cannot give that up, so without any good reason in your eyes, they reject your argument.
Just like you refuse to accept their arguments, because their arguments are based on fundamentals you cannot believe in.