If you like a brighter more metallic/less woody tone to your bass then the Badass delivers that due to its big metallic weight. String vibration is reflected rather than transferred into the body, hence a brighter tone (Listen to most of Mr Lee's recordings - usually very bright trebly bass sound).
But if you want or like a more trad Fender sound, go for the stock type bridge - cheap thru many many ebayers and the Badass holes are exact same as stock screw holes(Mr Badass made sure his bridges were an easy swap onto Fenders)...
Lots of people love the Badass, but it definitely alters your Fender bass away from its original warmer woody tone - it's down to your ears, but of course, your eyes come into it too! I agree that BA's are a serious affront to the beautiful flowing aesthetics of a Fender Bass, and Leo's minimal size and mass bass bridge was designed to allow tonal transfer thru the body, and to look just right with the rest of the bass's design(and then he stuck a big ol' ashtray over it!)

And the blond signature Geddy Lee neck being all maple gives a slightly more bright tone to the overall bass sound too...rosewood fretboards will slightly darken, or mellow your bass tone...these effects are kinda subtle, but if you're into the small things...