Will- unless I misunderstood, this guy is looking for gear for a turkey SHOOT, not for turkey HUNTING.
Barrel length will not be as important as choke. For most of these, shooters are going for extra full (a non standard choke that holds a REALLY tight pattern).
I have one gun I have done a few shoots with- H&R 348 Gamester- with a 32 inch full choke bbl. But back in the day I watched my Dad win 7 of the 9 matches at a turkey shoot with a Fox Sterlingworth side by side. Dad knew where the pattern for that gun fell.
Patten whatever gun you plan to use with the shells they use (commonly low brass field loads, fine shot) and do so at the distance they shoot (around here, 20 yards) You DO need to aim, not just point. So know exactly where you should be aiming on that little card.