Just a couple of questions... what are you using for and audio interface and is the bass plugged directly into the interface or are you coming out of some kind of preamp. Does the bass have active electronics?
Also.. if you're recording at 24-bit you don't have to worry about recording as hot as possible so you can give yourself some headroom and tame the low E string with some EQ.
Gain staging is important.. if your bass has active electronics you will want to set the tone controls to a -flat- position and turn the volume down. The audio interface volume should be higher than the bass volume. I usually set an interface input volume to about 7 then bring the instrument volume up until I get an acceptable level.
Since you mentioned the E string is loud on any fret...... does the bass have an adjustable pickup, can you bring the pickup height down a bit on the E-string.
Bass ----->--<--Interface Input -----><--Logic Effects----->
Gain staging is the proper setup of volume so that that an input stage is not overdriven by the previous output stage.