The whole point of having an active pick up IS sensitivity,
it's sensitive enough to give you a big overdrive from an average note, sensitive enough to reproduce the difference between a quiet note and a LOUD one. If you want an insensitive pick up buy something passive and cheaper!

I use an EMG 89 on my bridge, it's splitable and active. I haven't tried my guitar with Di Marzios but I hope that mine's just as sensitive cause it cost me £70. It sounds great clean but can't comment on how it sounds with distortion (my pedal's buggered, but, when it's fixed shrapnel's gonna fly!). I expect it'll kick ass but much of a guitar's tone lives in the amp anyway. Cheap guitars can sound glorious with the right amp and effects. If you don't have a good amp you might be wasting money on a great pick up but if you want, eventually, to have a top out-lay anyway though you've got to start somewhere and starting with a pick up's cheaper than starting with the amp! Besides, it'll benefit your sound whoever's amp you use.

Other than cleaning up your technique (and for all I know you might be a hugely radical player already) here are some suggestions for changing the sensitivity of your equipment.

SPLIT YOUR PICKUP
When my pickup is split it's also passive so the volume's much less giving me some control over its sensitivity. This reduces the volume and gain so in effect it's like having a lead and rhythm setting. This uses something which I'm told is called a "coil-tap", on my guitar you pull the volume knob up a bit and it goes single-coiled.
It's possible to buy extra boosters which puts up the output of the single-coil setting but I'm not sure what the point is, I like the versatillity.

USE A GATE THRESHOLD
My pedal includes a noise gate with different threshold levels. When the note falls below a certain volume it cuts out ENTIRELY. This is potentially very useful for playing crazy tapping stuff where you don't have the chance to dampen every string. It also cuts out some pick attack noise. On the other hand it's a pain in the arse if you want to sustain a note and let it die naturally and it just cuts out on you, also it rules out volume swells which are a big part of my style. Bear in mind that buying a separate gate threshold pedal would introduce extra wires and contacts and add his to your tone, it's best to use one built into a multi-fx pedal or amp.

CUT YOUR COMPRESSION
I like compression myself but if you've got tons of pick attack noise you might want to ditch it.

STRING DAMPENERS
It might be worth getting a string dampener, it's like a capo that doesn't work properly! It's nothing to be ashamed of (even the great Jennifer Batten uses one). Failing that tie a sock or bandana round your fretboard, might be useful for tapping but it's sad to loose the option of open strings.

ONE OTHER THING
You might consider is getting a guitar tech to stick an on off switch on your pick up. I did for mine, so now I can get these cool stuttering effects and switch squeals and bends on and off so you only have to strike the string once and get less pick attack noise. It cost me but man is it cool!