I do have a fretless SB-2 and love it, but I'm not sure it would be what you are looking for.
lighter & a smaller neck? probably lighter, definitely smaller neck
bridge pickup? Yes, but. And it's the "but" that may the problem for you. IMO the SB-2 bridge pickup is just an adjunct to the P pickup that is used for adding some texture to the sound. It's overwhelmed by the P pickup in normal settings so it's not really a traditional P-J setup. I know of at least one person who cranks the P pickup all the way down into the cavity and raises the J pickup as much as possible to get a better balance (I've never tried it so I can't say if it's fully balanced). And it's an unmatched single coil pickup, so I'd guess if noise was a problem for you with your L1K that the bridge pickup on an SB-2 is not going to be any better.
If I were a bridge pickup guy I'd probably want an L2K with a skinny neck for fretless (but I'm a neck pickup guy so my SB-2 fretless is wonderful).
Here's an SB-2 with its pickguard off:
this was an older bass that had a diamond plate pickguard on it (hence the doubling of the pickguard screw holes in some places). It doesn't look really set up for losing the pickguard to me which is a shame because it covers a lot of nice wood grain on a translucent finish.
If you are considering the trans green one thats floating around it's a good bass. I had it for a while (decided I don't play fretless well enough to keep two around) and other than a couple of fretboard marks it was super.