It would be amusing to think there was a similar pronunciation debate with Amwythig as the Shrow.../Shrew... divide with Shrewsbury.
I'm from Shrewsbury, and locals say it 'Shoes-bury', so there are actually three pronunciations!
I normally hear "shrowsbree". BTW Shakespeare would have pronounced his play, the taming of the 'shrow' (rhyming with 'blow').
I was traveling in the UK with my home base at a friend's house in London. Every night I would call her to tell her where I was. How I pronounced the towns let her know that I was really there!
Perhaps you could argue about whether the "w" is consonantal or not :)
"Am-wy-thig" (consonantal W, vocalic Y) or "A-mwy-thig" (vocalic W + vocalic Y diphthong)
Why isn't it "dw i'n gweithio mewn Amwythig"? Is yn specifically for proper nouns?
It's more that mewn is used for indefinite nouns, i.e. where you would say "in a ..." (e.g. mewn siop "in a shop").
But proper nouns are definite pretty much by definition (there's only one Shrewsbury).
You would also use yn with a common noun that is definite, e.g. "in the school (= the one that we had been talking about earlier)" as opposed to "in a school (= that had not been mentioned before)".