LP's aren't built for fender necks either. and if it measures half of the scale length at the twelfth fret, it doesnt matter what it was built for.
id assume youve been truly anal about tuning first of all. dont try to tune the guitar by fretting, to be sure, use a tuner, or the tuner at 8notes.com or something. as a compromise, tune with harmonics, but dont fret the strings. check intonation by playing the 1st (12th fret) harmonic, and then fretting at the 12th fret, they should sound the same. good luck.
also, if you have a great ear, it could be the intonation system getting at you. however, if its just this guitar, thats problably not the case.
