There are so many variables that make up (and can dramatically change) the sound of a bass that the possibilities are kind of limitless. Electronics, construction design, woods, finishes, etc. All may account for big or small influences on your sound. I struggle often with whether to tinker or not. In a way, one of the things that's led me to do more of my own building (more like, a little building and a lot of choosing and assembling parts) is the desire to fine-tune my way to a more likable sound.
One thing going for you with your KS is that you sound like you really like its feel and playing sensations. Frankly, if I had your bass and felt that way, I'd be looking for non-surgical (i.e. no routing) mods to the electronics (hands-down, the least intrusive mod) to see if I could get to where I wanted to be (or, just as likely, someplace else good that I never thought of before).
The hard part is that it's hard to guess what would be best to try - basses and builders are so individual, so varied and feedback from others on specific changes is so hard to match with your particular design issue (and so subjective, too) that you're sort of on your own. But I think for some of us, sometimes it's worth a try anyway. Basses that meet the criteria your KS has already hit upon don't come along every day. Maybe worth the risk/cost of trying? If you were successful, wouldn't you be glad you did?
(I can't believe I just posted encouragement to modify a designer bass on this Club...Well, at least I didn't suggest rethinking anything about an Alembic (gasp) - LOL)
Bill