How long has it been since you snugged the neck and bridge screws?
+1
Good start is to check bass acoustics without pluging it in. I hold it upright like a wall hanger (close to nut) with one hand, and strum it with other hand, first all strings, than one at the time. Listen for sustain, clarity, possible ratling. Do all strings sustain same lenght of time? This eliminates electronics, and tells what to expect from instrument when plugged in.
If it has poor sustain or clarity, check neck-body, bridge-body and nut-neck connections. If it sustains nice, check electronics, but also ALL of your signal path.
As for bodytrough vs. bridge-top string load, and also lightweight vs. heavy bridge: if strings and bridge are properly installed, there is minimal difference IMO. Maybe can be heard in perfect studio conditions, but not in real life, live gig situation, well - at least for me.