'68 383 barracuda fastback - what to look for?

Powerflite said

Correct me if I am wrong, but I had heard that pre-'73 A-bodies came with an 8 3/4 with big blocks and small blocks with manual transmissions. They also came in small block auto cars with the heavy duty suspension package. Otherwise, it would have been the 7 1/4.

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You're 100% right about the big block cars having 8 3/4s and all 340 cars have them as well.Other small block 4 speeds are hit and miss,I've seen 273 and 318 4 speeds with either a 7 1/4 or an 8 3/4.
I don't think the heavy duty suspension option would give you an 8 3/4 but ordering the trailer tow package would have.
That said,I've seen 8 3/4s show up randomly in low option 318 or 273 automatic cars but those are probably due to parts shortages at the plant.