If you get a tube screamer...you'll need a soldering iron and some cap / resistor swap outs on the output filter. The Tube Screamer design has a High pass filter which is cuts at a bass sucky frequency of around 160hz. On a guitar it helps to tighten and brighten up the tone, but on bass...say good bye to your bass boom / strength. The other issue with the screamer, is that the stock op-amp isn't that great and there are loads of options with the diodes. So I personally would look up a custom Screamer (there are lots about) with some custom switching. I would choose one with an OPA 2134 Op-amp. I have two tube screamers, one is a bufferless Clay Jones clone, which doesn't have any output HP filter at all. The other is Screamer clone with an active Bass / Treble / Mid and mid freq tone controls. Out of the two, I prefer the Clay Jones. But in my opinion there are a lot of better choices for bass friendly overdrives out there. Some even work well with guitar and bass. Overdrives like the OCD, Barbershop, mojo mojo, Wampler Ecstasy, Timmy (very nice) and the list goes on.
There are other overdrives which emulate specific tube amps, like the Catalin SFT (like a farty SVT), Joyo American Sound (very nice) and again, the list goes on.