Yes, this is normal for a pickup sitting so close to the strings. It is similar with some bridge wing pickups.
The high frequencies will be more damped towards the bridge feet or the top.
You might want to use a small clamp (or two) tepmporary to hear if the friction noise will be reduced that way and decide yourself if you might want to get the regular clamps for it. (A higher additional mass on the bridge will change the sound, so concentrate on te friction noise, not on the overall sound when using a higher mass clamp.)
Without a very high impedance amp or preamp input (about 10 MOhms) the lower frequencies are damped (specially if it gets louder). For the price of a preamp you might get a better pickup instead that works without a preamp with your amp.
An read about the BP 100 here on talkbass. Only some basses work well with it.