I just spent some analytical time with my Model 1996, and for the most part, I can't offer anything new, really. That B string is unwieldly. I didn't really notice an overtone, as a sound/note without any sort of buzz. My feeling is that the quality of the note is largely dependent of the fretting. I've been playing my acoustic mostly lately; Taylor GS mini. It sure seems a lot easier to play/fret than my 1996. Unless I fret really close to the fret I'm going to get some sort of odd sound, depending on the exact fingertip location and/or the fretting pressure applied. So I have to fret close to the fret and use a considerable amount of finger/fretting pressure; those few frets closest to the nut require both; there's only one long and stretchy string side way up there (vs. a good length of string on both sides of the fretted fret halfway up the neck). Those very small frets of the 1996 demand close finger placement/fretting to the fret. I initially suspected there is less of a break angle at the first few frets, but after further thought, with proper fretting those strings breaking over the first few frets closest to the nut would seem to have the greatest break angle compared, say, to strings-frets closer to the bridge; it's just tougher to get that at those first few frets. As I move fret-by-fret towards the body of the guitar, both finger pressure and precise fretting become less demanding; any string is easier to fret at the 12th fret than it is at the first fret, and no fret buzz.
The frets on my acoustic are about the same size as the ones on the 1996, maybe slightly bigger/taller. I spent a lot of time getting a good, pretty low action on that guitar. Maybe I should lower the bridge on the 1996. (It should be easier to play than an acoustic.) If I could avoid out-and-out fret buzz, it would probably just as easy to play as the acoustic.
If this "ping" overtone were present only on the open, non-fretted string, I'd say that the nut slot would have to be refined (proper slope). In my case, I think it's my failure to fret properly. Sometimes you just can't fret all strings properly at once -- I do not have any crooked finger joints at all, and for, say, a cowboy "A" chord I have to use three fingertips; not really that easy on a little Rick fretboard if you're really moving along with your tune, compared, say, to a big Fender neck.
One last (final?) option would be to deliberately change the tuning on that one troublesome string -- either decrease or decrease the pitch of the open string when you tune up. You'd have to experiment in both directions. See what it sounds like. I doubt some minor off-key discrepancy would be very noticeable, at least when playing chords.
If you were to use the same gauge string as your e string, and tune properly to pitch, that "B" string would be easier to fret and maybe, if there is a fretting issue, this would be resolved. I don't know. Maybe just ignoring that "12-gauge" string recommendation (requirement?) and going with a lighter set of strings might make things easier; a set of "11s" maybe.
Well, all these thoughts are pretty much based on my critical re-visiting my 1996 tonight....
I'd like to hear some other opinions. A few years ago there used to be a lot of 325 discussion here on all sorts of topics. I think a lot of guys went and bought bass guitars.