I never know how quite to respond to very rigid thinking without offending said other. I can explain that there are other schools of thought. Not sure really if its even my business to bring it up.
Has anybody else come up with ways to discuss issues in a non-threatening manner that doesn't upset this type of thinker?
Black and White thinking (and feeling) is part of autism. There are not a single autists who is not affected by this one way or another.
The way I have worked with my father and with myself is to force greys. Lets illustrate it with an example:
Lets say that when I was a kid, I met another kid at the shool and liked him. As I was affected by B&W thinking I did not considered him a classmate, but a SUPER GOOD FRIEND!!
Why? Because I lacked the intermediate categories that go from HATED ENEMY to SUPER GOOD FRIEND. There was no "Classmate" category, no "guy who gave me good first impression" category, no other category.
This is something that happens to all human kids, but as NT kids grow they create all kind of intermediate categories with all the correspondent body languages, reactions, what to share, social rules, etc... Autists dont. Some autists will create some of the many possible categories, but will never achieve the flexibility NT adults have to jump from one category to other.
So the way to combat B&W thinking is to create those categories in a purposed artificial way. So I need to create the "classmate" category and learn what are their characteristics, social rules, etc. The same with the "coworker", "boss", "teacher"... Then add labels like "religious person", "sensitive", "rigid" to know what topics to avoid or how to better adapt myself to them.
So once I have done that, my thinking is no longer SO black and white as it was. It is more flexible now. On that topic.
For every topic there is "grey" to build to fill the gap between black and white. So to help another person, you can guide them to create new categories that are in the grey zone. To decide how would it be appropiate to behave in that "not white/not black" situation.
Im not sure if I could explain myself, let me know if I missed something.