Back to the topic at hand, the noise gate seems to be the only thing that really brings the noise down but I have to really crank it. The next time I play out with it, I'll try setting it pretty high to see if it's taking too much of the initial attack with it.
Definition on the amp didn't do much and pick attack did nothing. Interestingly, definition in the Kemper Drive did a lot as did bringing down the tone, but that just made the overall sound too muddy.
I'll also look into that fast fret stuff.