Picks and technique can make a big difference, but you say you're not getting this tone from your other guitars. This makes me thing the guitar is the culprit.
Does it have a tone control. Some '80s shred machines took the minimalist approach to wiring.
I had a strat style with a SD JB humbucker that had a similar problem. I think this was caused partially by the wiring. It had no tone control at all. A passive tone control will bleed some treble from your signal, even when turned up all the way. That's why strat bridge PUs can sound so damn screechy sometimes - no tone control. Passive tone controls are very effective at getting rid of the ice-pick sound.
If it has no tone control, you could ad a micro-pot inside the cavity, or add some kind of stacked pot to the existing volume control, so you don't have to drill any holes in the front.
If it does have a tone control, you could try changing the capacitor. Typical values range from 0.001 to 0.01. Increasing the value will roll off more highs.
Also, the value of the post can effect tone. A 250K pot will give a warmer tone than a 500K. You could try this.
I suppose changing the pickup could be considered too.
If wiring or PU changes make no difference, and you want to keep this guitar, you could try experamenting with a parametric EQ. These will allow you to shape, or eliminate specific frequencies - can be had for around $50.
-Sven