Okay - this is an oldish thread and my experience with a LPC is from the mid-70s, so not sure how relevant my comments are. But just in case . . .
I bought a used LPC Black Beauty "Fretless wonder" in 1975 (or was it 74?). I think that the LPC was a 72, but I'm not positive about the year. I know that it was built in the 70s. That guitar was HEAVY. I never measured the weight, but my recollection is that it was well over 10 lbs. I never thought that it sounded any good, even though I replaced the stock bridge pickup with a DiMarzio Super Distortion.
One day, after I had been practicing, I wanted to go out, so I left the guitar leaning up against my amp. When I came home later that evening the head stock had snapped off the neck. I had never dropped the guitar or bumped the head stock. The finish showed no signs of having been previously damaged or broken. The head stock was completely broken off. The break ran along the grain of the volute rise. I got it "fixed", but the guy that did the worked really sucked and it was never right. I eventually sold it for a pittance due to the condition.
The only thing I liked about the guitar was the action. It was effortless to play. But it was so heavy that I never really enjoyed strapping it on to play it. Around this time, after the head stock was "fixed", I broke my thumb. I completely trashed the back of the neck because I kept playing the guitar even though I was wearing a cast. If that guitar still lives, I can only guess the stories that have been made up to explain all the damage.
Clearly there is no disputing that construction techniques have a huge impact on tone. That being said, I have 4 Knaggs guitars that are as good sounding (or better) as any Les Paul that I've ever played, with the possible exception of a 59 sunburst that I got to play when I visited GIT in 1980. I only played that guitar for 5 minutes or so (it belonged to a student who was giving me a tour of the school), but it sounded magical. At least that's what my addled memory tells me.
The Knaggs Kenai J that I bought used for $1885 is an outstanding guitar. So is the Knaggs Kenai (no J) that I own. Both are roughly 7lbs, 8ozs (according to the build sheet), have large neck profiles, and sound killer, each with their own voice. It's true that they aren't Les Pauls. But I'd put my Kenai J up against any Les Paul flat top. You might like the LP better, but not because the Knaggs is deficient. It would just be a matter of taste. IMHO, of course. YMMV