Hang on there - the active pickups might need adjustment in height and pole piece stagger, but I think it really requires passive PAF style pickups, end of story. And, do not give up the fixed bridge idea. One does need that. Does not have the same "vibe" without the "trem springs", but I enjoy both.
I did read somewhere on the interwebs about a signature player that went back to passive pickups after playing actives for years, saying the actives do not have the sound. David Gilmour? Your JS is a "hands on" guitar, I suspect if you get the Cort back to "traditional" (no such thing), you will want to keep it.
This is a nice guitar, sort it out, even if it means having to spend money (next year) on lower output PAF style pickups (and the right pots and caps...). A LP style will be a LP style, shape, feel, weight, neck profile wise, I suspect (never even handled one), and if you have a good guitar, if it plays lekker (after adjusting from the JS), if you look at it and think "yes", then sort it out as you go along. The other option is a fixed bridge Ibanez with a neck similar to the JS100, with good pickups. But, then, that would be just an Ibanez. A very nice one, to be sure.... In fact I need one of those, with a neck profile like my Jem Jr. My current fixed bridge needs a wider neck. Any old excuse, eh? As for mud, after my "tone pot and treble bleed mods" thing, I cannot complain about mud. Boy, is it bright now... I can fix that, and I now know that guitar does not need "better" pickups to get what I wan, sound wise. My Pointy toy, well, it will have to endure a pickup swop. Which will bring it to less than half the cost of a new "upgrade" guitar.
The advice from studmissile is good - play it for a while. Then work on the pickup thing (are the caps the correct value, does it have 25K pots?). Sigh. Might need more money. Is it wise to spend money on it? If it plays, if it feels good, if you look at it and think..., yes, for sure. Normal advice to rather save up for a real guitar to be ignored in that case. Pickups and pots for say, R2 800 (unless you go Duncan), might be better than another R10 000 to buy a "better" instrument, which also might not be "quite there".
An aside, the photo of the amp shows a "shark tooth" inlay Ibby in the background. Yours?