True bypass is only impacts the signal chain when the pedal is not engaged.... and if done correctly, it will not impact a passive pickup at all. Of course, when you engage a pedal, the sound will change, due to the buffering of a passive instrument and the nature of what the pedal does. Might just be some strange impedance mismatch or something with that particular pedal and the pickup.
If you use a compressor at a very subtle setting, getting a tube preamp does about the same thing. My Genz Streamliner, which has a true 'high voltage tube preamp' section (versus the more 'marketing tube' executions in many hybrids) really does a great job of subtle, musical compression when you dig in with a P, and no pedals or multiple cables needed.
If you still would rather use on outboard compressor, I would suggest heading to Guitar Center or whatever with you bass, and seeing if there is something in the design of your current unit is causing the issue, or if it is an outboard effect thing in general that you are hearing.
As a rule, I hate ANYTHING except a true bypass tuner between a passive instrument and an amp, and I go so far as to use super low capacitance cable with my passive P (George L's) to make sure there is a minimum of top end attenuation.
Edit: Awesome bass by the way!