If you're dropping 5 grand on a guitar, you definitely should be able to figure this out without asking NS...and what do you mean by "pulse as opposed to just running on?" Do you mean you want a more "raw" feeling guitar? We need more info.
As for buying an electric, I will say that you shouldn't buy one unless you have played it in person. That is the golden rule of smart guitar shopping. If you have a Les Paul, I'd imagine you are looking for something other than heavy humbucker sounds, correct? Have you looked into a quality Strat? Jazzmaster? Baritone?
A good way to check the acoustics of an electric is to hold the butt of the body (by the strap button) and plug the G string. Try this on various guitars and compare how long they sustain. Then go around playing as many as you an unplugged, getting a feel for the acoustics, dynamics, and feel. Then, play them all through an amp similar to yours, identical if possible.
IMO nothing beats a good Strat. They can sound just as fat as LPs but with more dynamic clarity. Also, they're fucking bombproof. You can drop them and smash them up and they will still work. If you have 5 grand to spend, perhaps look at a custom? Check out www.warmoth.com.
They make amazing guitars and have excellent QC.