Did the guy at the store tell you the Badass bridge was brass? It's not. It's always been a diecast zinc alloy. I don't know how that rumor ever got started. It does have more mass than most stock bridges and will probably affect the sound to some extent, how much depends on a lot of factors.
Nut material does affect the sound on all strings, not just the open strings. A brass nut and brass bridge will act as an inertia block, keeping more string energy in the strings instead of going into the wood where it affects tone in a different way. Your sound will have more sustain but less of the sound of wood.
Brass bridges and nuts are out of favor in the bass world. Some people do still like it, and if you really want it, that's fine, but don't let someone else talk you into that sound.