I'm considering the possibility it's just a humorous and slightly absurdist excuse to not remove the hat. Or an inside joke. The real reason might be more convoluted and/or personal, like having a gorbachevesque birthmark the person is not entirely comfortable with or whatever.

If it's not a joke/excuse as another comment mentioned, it might be connected to how some people sneeze when they are exposed to sunlight. Maybe it started like that for him, and then became associated with taking the hat off enough to trigger it inside as well.

I've got the same sneeze reflex https://en.wikipedia.org/wiki/Photic_sneeze_reflex