Well, when you compare the two, you have almost the same rifle. The LRH has a 26" CrMo barrel, an adjustable cheek piece, and weighs *almost* a pound more. The BH has a shorter 23" barrel which is SS and fluted, I'm guessing those two things make up the majority of the nearly 1lb weight savings, and no adjustable cheek piece. How do you intend to take the majority of your shots? If you plan on being in the prone position more often and intend to run a bipod, the LRH hands down. But if you plan, or expect, to shoot off the shoulder from standing/sitting/kneeling positions, the BH may be more manageable. In my eyes, that is the main trade-off between the two. Regarding ballistics, some people consider a 26" barrel the practical minimum for a .300wm, especially if you are shooting long range, and on paper, the 26" barrel will outperform the 23" with higher velocities and flatter trajectories. In practice, at the range you specified of 1000 yards, most optics that would be a reasonable match for either rifle will have more than enough internal elevation to keep you on target without hold over.
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