....and the things you mention primarily (excepting only the pickups) have to do with the way the guitar feels, not sounds.....
But tone is definitely subjective, so I do agree to disagree. The SC245 is certainly a fine instrument.
Another vote for the Cobra S by Tom Anderson. Had one and now regret ever getting rid of it.
Try a 335.
Custom strat, mahogany body/neck, rosewood fretboard and a pair of PAF style pickups. I'd probably go hardtail on it too since you'll be calling the shots. I don't know of anything off the shelf that will work. I'm in the same boat and have been looking into something similar.
A massive part of the Les Paul sound is the combination of a fixed bridge and the 24,75" scale length. [...] you might consider buying a conversion neck from Warmoth [...]
To look at, far, far, away from a Strat in feel and way awkward, close enough to the LP in tone.