I see two possibilities:
1. Their pronunciation of Hanzo and Schultz is intentional
Budd may have ended up a dead-beat, alcoholic bouncer at a titty bar, but back in the day he was part of the cosmopolitan Deadly Vipers, along with Beatrix (fluent in Japanese), Bill (who can presumably speak Cantonese and Spanish) and the multicultural O-Ren and Sophie. His pronunciation of Hanzo and Schultz is actually the correct pronunciation in Japanese and German, respectively. His adoption of the bushwacking Texan persona we see is totally cynical (i.e. purposeful), as he refuses to take himself seriously.
Elle, on the other hand, is presented as a poser. She is attracted to the power and the glamour of the assasin lifestyle, but lacks character and patience. Her interest in the Way of the Warrior (Bushido in Japanese... not sure what Pai Mei would call it) is akin to that of a fan girl/boy, and not a deep, spiritual commitment like Beatrix and O-ren demonstrate.
So what I'm saying is, Budd picked up some culture in his travels and Elle is the real rube in that trailer.
2. Only Elle's pronunciation is acted, while Budd's is a mistake by the actor (Michael Madsen)
Because of the above, it was in-character for Daryl Hannah to butcher the name of a legendary swordsman and to say Schultz with an American pronunciation. Elle just doesn't know any better.
But Michael Madsen is actually into writing poetry. In my experience, people who enjoy language know and prefer original pronunciations. So maybe this was just Michael messing up his Texas accent?
I vote for option 1.