As far as choosing a range that is best for jazz (Bb or G vs. C or A), that's an interesting question. I play the key of A most commonly, and I rarely find myself wishing I had lower notes. The one time I might miss lower notes would be if I'm learning a solo off a record, but when it comes time to play live, the key of A seems satisfactory 99% of the time. I think I would very much miss the A, Bb, B if I switched to the key of C, however. Bottom line, I don't think you can go wrong with a Bb or a G.
As far as enharmonics that are best for jazz, my take is that the keys that have a small advantage - due to enharmonics and the combination of the 2 basic dimi scales (Whole/Half and Half/Whole) - on a key of G or Bb dimi would be:
D, F, Ab and B Major (including ii-V-Is).
D-, F-, Ab- and B- (including iim7b5, V7, i)
G, Bb, Db or E pentatonics, minor pentatonics and blues scales are convenient.
basic bluesy patterns work well over C7, Eb7, Gb7, and A7 chords
Those same keys on an A or C dimi would be:
Db, E, G, and Bb Major.
Db-, E-, G-, and Bb-
A, C, Eb and Gb pentatonics, minor pentatonics, blues scales
F7, Ab7, B7, and D7
Anyway, this is just my opinion, I'm sure you have your own and will discover plenty more.
in the case of the Suzuki, A or C since thats what's available although, I'm not sure if in this case, it would be better to tune up or down?
If you wanted a Bb dimi, then I think you'd want to start with A plates as opposed to C. I usually take the route of retuning from A Maj to A dimi (same key plates), but have heard that Gnarly has done half-step-up retuning to dimi.
Maybe Gnarly will confirm the half step up tuning...