I don’t have a good answer, but I do have an idea. The next time you face this situation, instead of jumping to give information that you would need to know in order for you to believe what you are saying, ask the other person instead: “What would need to be true in order for you to believe X?”

While I’ve often found that people can’t handle this question and just disengage, if you do get an answer, then you can actually try to say what they need to hear. They may move the goalposts after getting an answer, but it’s better than wasting time throwing shit at the wall to see what sticks.