intonation madness

Is this similar to the Earvana nut?  Those are supposed to be great.

how is the relief on the neck? this can be a big factor at times.  . . if you cange it via the truss rod. . . you will have to re-intonate as well. . .
how is the nut height?  . . . sometimes too high can cause things played in the first frets to be too sharp. . . . on otherwise intonated strings. . .
it sounds like you have a lot of experience with this. . .  probably much more than me. . .

. . . if it's 'above' you. . . take it to someone who you trust. . .

You stated, "I make sure the harmonic at the 12th fret is in tune," which is not the  important first step . The first step is making sure the open string is in tune. Then plunk the 12th fret harmonic and remember it's reading on the tuner, then fret/play the note at the 12th fret ,and  check your new tuner reading for the difference between the two readings. It is from here you make your saddle adjustment....."Flat forward, sharp back," I believe is how I remember it.
The adjustment is the difference between the harmonic and fretted note at the 12th. As you know after a saddle adjustment, retune the open string again, and start the process over.
I use a Boss tuner with the needle cause I can see it flutter, and can tell how much incremental saddle adjustment to make. The digital flow type tuner isnt sensitive enough.
One thing is to check the tuner immediately the instant you pluck a string. Letting a string ring out eventually throws the tuner into displaying note degrading overtones.
I respect your efforts to seek this help. Back in the day when I was 17, I struggled with your same problem, until I learned about intonation.
The only other idea here, may be pup height. Are the magnets pulling to hard on the string, messing with it's normal eliptical vibration? Or are the pups to low?
Other idea....do the strings run straight up the neck? Is the neck off that way?
Only have 3 wraps of string around the tuning posts, no more, or else you get loosening. As mentioned above, check the nut for pinch.
Obviously, an intonated guitar starts with the pegs, down to the stud. Good luck, you seem to be more experienced than me, & i'm just trouble shootin'.Thanks >>DAN

That's cool. My point above is to not try to initially tune using the harmonic. Go by the difference between the harmonic and fretted 12th fret note.....nuf said.
Sometimes, these things just "play themselves out," like a fret buzzing all of a sudden....could be due to humidity change....give it time. ?
I saw in your photos that the stud is not screwed hard and fast down to the top of the body. That's fine. Just make sure that the string angle from the stud to the bridge isnt so steep, that the string leaving the stud is touching/resting on the back edge of the bridge before it rests in the saddle. (I found this out not to long ago on mine!)First point of contact, obviously, should be the saddle only...... getting too picky perhaps, but just throwing that out since it's a new guitar.Run a piece of dental floss, or thread, under the string at this point to make sure the string isnt touching the back edge of the bridge.
Factory set ups are not personalized. I just hope it all works for you. I think the Traditional Pro's are a good price, and it is a L Paul! My 89' LP cost $1,200.00
Only other suggestion would be to adjust one thing at a time. My strat input started crackling, so just tightening the nut seemed to solve the "problem," we'll see. >> DAN

Take a look at some of the guitars here, to see how some makers have tried to solve the general intonation challenge: http://www.microtonalguitar.com/