The question keeps coming back. The Conn players answering and the answers stay the same. 😉
I'll add my 2 cents/pence.
I own and play a 1936 12M (I know that is 20 years younger), which intonates perfectly (I mean really perfect!) with a "pickle-barrel" style mouthpiece. I have a copy that Clark Fobes makes or used to make. I added some baffle and eventually had Phil Engleman improve the baffle and make it permanent and also put another facing on it. It was kind of a detour, but the piece has some edge now added on top of this big sound you can't find on a modern setup, it has gained lots of power and is definitely the best piece I have ever played on an old Conn with original crook. Unfortunately it is unique, so it doesn't help anyone if I tell about it, unless you want to go a similar way.
All other pieces just don't work the same intonationwise for me. Long middle D and E are always sharp. With "normal" large chamber pieces like Links or the PPT it is certainly in a range that can be handled by a good player, but it's still not easy for me. What finally did the trick for me for more modern pieces is the Wilmington 12.5M neck by Music Medic. It is not quite cheap, but an impressive demonstration by someone who understands the "missing cone" thing. I can play any modern piece from paint peelers to "normal" large chambers with it easily on the 12.5M, the intonation is just right. I love the PPT, but having both a unaltered original neck and the 12.5M neck, I prefer to play it on the latter.
A neck extension is a "classic" option for the problem, almost every other old 12M I have seen has had the neck extended. But it changes originality and will likely not be perfect for reasons @turf explained.
If you are willing to play around and do some research, you will find an article about "tuning a sax" on the music medic blog which descriptions how you can tune the second octave on a sax and another blog somewhere where someone fixed 2nd octave intonation of a Martin baritone by soldering a piece of thread in the pig tail. I have played around with tubes in the crook and pearls on a wire and find it quite amazing, how intonation can be "tuned" for single notes. However, I never made permanent changes myself in this regard, fearing it would affect something negatively. But if you are good at this kind of work, I am sure you can tune the second octave to fit a modern piece on your horn...
Key heights are another nice rabbit hole to get lost in. But it is a different rabbit hole. Key heights will affect both the 1st and 2nd octave in a similar way. This is not your problem, although it might help with a compromise. Nevertheless, the key heights should be right in the first place, before you start messing with other stuff.