Hi Andy, same issue I have had on my unfinished Tele.
When using a chromatic electronic tuner to tune up, playing root chords sound ok, particularly D but when checking across the strings at fifth fret or 5th to 7th harmonics EAD are OK, G sounds a bit off, B sounds even more off and this results in high E ending up sharp when re-tuning the pitch by ear.
This Tele has had 3 nuts and 3 bridges on it plus I have lowered the neck pocket height to bring the action down and that seems to have done the trick. Now strung with 10-46 and been playing it on & off for past month or so to see if it keeps behaving itself. So far so good but having mostly played 9-42's the finger tips get a bit tender.
So how does this relate to your predicament? TOM Bridges don't have much scope for adjustment therefore you probably already got that part sorted and have that lovely spalted LP as a base reference point to work off. This leads me to the next part in that maybe checking how the LP has been setup and then replicating same on the 6 string neck might be the quickest way to getting it sorted. Definitely sounds like action is riding a bit high somewhere and nut would seem to be the most likely culprit but it may be something else. How is the neck angle on the 12 stringer side of things? Are they both the same? Also how deep do they both sit in their neck pockets. With this axe being a bolt-on neck design there are many parts that could need some fine adjustment.
Hope you get it sorted without too much grief and frustration.