Here's a possible approach to the problem:
Determine what bullet(s) you would want to hit with and have expand at 200yd. Find out from the bullet maker what velocity is needed for expansion (for example: Sierra 125gr pro hunter needs 1700fps; Nosler 125gr ballistic tip needs 1800fps.) Now, determine approximate muzzle velocities obtainable with different length barrels and run a ballistics program to determine what velocity you will get at 200yd. If enough velocity to expand the bullet, you will have your answer. I use the Nosler 125gr for deer hunting. Even with a 16" bbl, the calculations show the bullet should only expand out to around 190yd, so 200yd is gonna be sorta "ify". Bottom line, if using something other than Barnes copper bullets designed for the 300 Blackout, I don't think you can get reliable expansion at 200yd from any short barrel 300 Blackout. Using the same Nosler 125gr bullet out of my SBR (9.5") my load is only good for expansion out to about 160yd due to less muzzle velocity from the shorter barrel.