There are actually many ways that a stereo bass can be run.
One common way is to run each pickup to a separate amp/speaker setup. One would think that the character of each amp would be part of the approach.
Another is similar to what you may be suggesting - run each pickup through its own signal chain, then blend them and play through one amp. This can be accomplished with an Alembic F-2B, which allows you to either run each pickup to separate amps or blended and sent to one amp. I have also successfully used a mixer to combine the signals.
One dream set-up is an SF-2 Superfilter and F-2B preamp - then you get to decide things like which one is first in the signal chain! Can also think about effects - overdrive/distortion on the bridge pickup, clean neck to preserve the bottom, octave only on the neck, clean or modulated bridge, the possibilities are endless!
Have fun experimenting!