As others have said, the right flats and some foam can always get you in the direction you want to go pretty quickly.
Just to clarify my earlier comment, I don't think the SB-1 would be a bad choice by any stretch, especially if you like the SB/Kilo body shape. But from my experience, which includes among others an L-1000, a couple of second-gen SB-2s with tone controls added, an LB-100, and an L-1500 (precursor to the Kilo), I have appreciated the SB-2s more as a modern P with rounds and my L-1000 just nails the tone I want from vintage thumpy flats. I do find that rolling off a little low end on the L-1000 and L-2000 helps with flats.
Edit: Oh, and the OMG on the L-1000 is instead of series. So you have parallel, single coil and OMG.