You could just get a 375 Ruger barrel. The Ruger case is shorter than the H&H and feeds perfectly out of a standard Win Mag magazine. But you can only fit two rounds in the mag plus one in the chamber if that's an issue.

As for the muzzle brake, if you can shoot your 338 with the brake off, then you shouldn't have any problems with the 375. The 375 H&H and 375 Ruger both have more of a push than the sharp jab of the 338 Win Mag IMO.

Andrew