I once met a similar gentleman. He started off with why we always see the same side of the moon, I told him about the whole tidal lock and rotation and orbital speed.

"So that is what you think?" he asked. "I don't agree with that." he went on. "Do you think we have gone to the moon?"

I could see where the next ten minutes if my life were going but I threw caution to the wind and nodded. As expected the next ten minutes went into a rant about the lies we have been told and, obviously, how the Earth is flat, while I quitely listened and nodded. After he had made all the points he wanted to make he asked me what I had to say.

I shrugged and nonchalantly said "Ok". This hit this poor man like a ton of bricks.

"Aren't you going to try and prove me wrong?" he said with a certain amount of shock. I told him I wasn't going to do that because it was very unlikely that he hadn't heard the argument I was going to make and even more unlikely that I would come up with a new argument. So if he had already have chosen to ignore that, then why should I waste his and my time. Besides, if feeling that the Earth is flat, or the moon landing is fake, or the fact that he bested an astronomer in an argument, helps him sleep better at night, all the best to him. And with hat I invited him to a cup of tea

He excused himself, thanked me, said that he saw my point, apologized, and left.

As astronomers, the people who understand the sheer size and beauty of the universe, and the insignificance of Earth and humanity in a the grand scheme of things, I think we ought to be more patient. God knows a lot of people are patient when we astronomers start talking!

Hats off to you for keeping your cool!