Different point of view
This issue probably has more to do with the barrel changing temp after the first shot...than anything else.
I really doubt fouling the barrel could have any predictalbe, much less positive outcome, and there is a way to prove this...read on.
Your particular barrel probably was not cut rifled, but rifled by another means(probably button rifling) and has some internal stresses induced by the process. When the temp rises, the barrel flexes enough to move your pattern, but once up to temp its good to go.
If your gun always shoots the first shot to roughly the same spot, say high and to the left. Your could verify this next time you go to the range and on the first shot lay the gun on its side. Sounds flaky, but try it, I bet the first shot is in the corresponting location 90? from the last time you shot.