There is no such thing as an all-in-one "Midi pickup", the type you are looking for is the "Hexaphonic pickup" (with 13 pin connector), which has a seperate pickup for each string.
The analog signal from this goes to one of two types of device.
1.) Pitch-to-Midi converter or midi guitar synth. A dedicated converter is the Roland GI-20, or one with a built in synth sound module like the Roland GR-20. These have issues with tracking delays which get worse on the lower notes.
2.) Modelling Synth such as the Roland V-Bass or
VB-99. These don't have the tracking delay found with pitch to midi (unless you are using the pitch to midi function in it) since they use the analog pitch as the synth tone generator (a hybrid system).
The only reliable "pitch to midi" or midibass system to date is a dedicated custom built bass using fret switches. Or a way around the tracking delay of pitch to midi is to use picollo bass strings, and tune the synth down an octave.
There are other options too, like a midi foot pedal note controller (like an organists foot keyboard), to run your synth sound modules.