Back in the 1950's Winchester's factory rep attended many shoots and hunts to demonstrate and advertise Winchester shotguns.
He had any Winchester he wanted, including a very high end fancy Winchester Model 21 double.
But for quail hunts he showed up with an ordinary battered Winchester Model 12 with an 18 inch barrel and open Cylinder Bore choke.
While everyone else had to wait for the birds to get far enough away the tight pattern wouldn't blow them to bits, he'd shoot the instant that they came up.
He always got his limit.
The full American Rifleman Dope Bag test used a Marlin Goose Gun with a 36 inch barrel.
The staff mounted an external choke on the gun, patterned it and did velocity tests.
Then they cut the barrel off one inch and re-patterned and velocity tested it.
They continued this until the barrel was at 12 inches.
Their results.......
Barrel length had no effect on pattern. Short barrels patterned the same as longer barrels.
Velocity was the same except when the barrel was over 28 inches, and then they saw a slight velocity loss due to friction.
Nothing "got out of hand" until the barrel was down to 12 inches.
Ballistically anything that was going to happen in a shotgun barrel happened by 18 inches.
A longer barrels sole advantage is that it points and swings better.
The idea that a longer barrel shotgun shot "harder" or farther was a left over from the black powder days when longer barrels did burn the powder more effectively.
Not being a turkey hunter I'd think a KS7 with a add-on tight choke and either a scope or electronic sight would work very well on toms.
The shorter barrel would probably make getting on target easier and without the long barrel moving alerting the bird.