I have the SR2405 with Aguilar pickups. To me it only is the D and B strings that are most noisy. I've had similar things with all of my basses(fender, squire, schecter, Warwick). I actively mute with my hands. If a passage runs across the strings I will use a floating thumb. I mean if you are trying to blast out some taps you gotta use both hands or just use a fret wrap. Keep in mind that's not my final tone with my set up either so that may change things. I have a HPF, LPF on the markbass(only turn it up a hair to clean up), compressor, harmonic booster, and B3K for dirt.
I could be wrong but I think part of a rich sound does this, and also I would never pluck a string and leave it the above open unless it was something like I was so focused on the lower strings.
Hope this helps some. Ideally a local store would have some stuff to try but I know that's not always the case.