Tube Screamers are very popular on bass even. The things to consider are that a Tubescreamer won't have as much distortion as I think your MXR Distortion will have, and that most Tubescreamers suck out your bottom end pretty bad. I would recommend looking at the Ibanez TS9DX, which has extra modes that are specifically designed to retain a lot more low end, which makes it a great pedal for bass, but if you want the TS-808 sound over the TS-9, there's also the Voodoo Lab Sparkle Drive, which is a TS-808 with a clean blend on it which allows you to retain your bottom end. I did some soundclips of that particular pedal a little while ago, check it out:
Voodoo Lab Sparkle Drive SOUNDCLIPS
Anyways, I'd say you're on a good track checking out the Tubescreamer circuit, it's capable of some cool sounds on bass.
