i have the same issue, i use a wood pick and it is worse! haha the wood just get torn away from the friction of the string, now i am using a dunlop medium and it is not that bad anymore,
try not use coated string as i realize it leaves more residue at the pickup, the coated will just fall off
one thing that i realise is that if i use light pick, i am able to get a much brighter tone noticeable in the high mid area of the bass than a thicker pick, but then your pick will break easily,
another way is to get that new cleaning tool from citymusic, cant remember the name