I've been playing distorted bass since way before it became popular. I am surprised you like the big muff on bass!? I found it to be anemic. I alsways liked the dod fx20 overdrive, the original rat, the turbo rat and the dual rat. Most distortions cut the low end pre-drive, to keep regular guitars from mudding out on chords. I have seen and used some modded overdrive/distortion boxes that have the highpass taken off. Les Claypool used to use a modified ADA preamp like this. As far as distortion pickups, pickups are like microphones and pretty much just provide the signal to the preamp to be amplified/overdriven/fuzzed etc. To get onboard fuzz you might be able to missmatch some pickups and preamps to get a fuzz happening, but it will probably be not too nice of a fuzz.
There were some basses made by Vox and also EKO, Teisco and other Japanese and Italian makers that had fuzzes and stuff built in. These are hard to find and with the ebay phenom, the prices are higher than they are worth, but that is probably your holy grail. Though most are hollowbodied so they feedback oddly. (I assume if you are into distortion you are into feedback and VOLUME).The Voxes are exceedingly expensive, they are usually teardrop shaped(think early Beatles) and have like 10 slide switches for all the pickups and effects.
On a side note I once chopped up a Fender P that most guys around here would kill for, so I could bolt a fuzz and a phaser to it and manipulate them while I played. Unfortunately my band at the time went hardcore punk the next month, and they jettisoned my weird self. Yeah, too weird for the punks, too punk for the artists... meh
. I just thought it would be cool to tweak your distorted sound from your bass. I know you can control your affeects from your bass, but thats a custom job I'm not willing to put money into... yet. 
