For a hunting rifle capable of taking game beyond 1,000 yards I really like to jump into the .338 caliber cartridges. I'm not sure why your concerned about firing more rounds per string while hunting, we all make one shot kills here right ;) A 2" BAT action with a 1.250" straight taper barrel 36" long will still need a barrel block. You can fit them into a McMillan 50 cal light gun stock and topped with a scope you'll still end up little over 25lbs. If you go to my website at this link Montour County Rifles
and look on the home page, upper left corner is a .338-.378 Weatherby on a 1.500" Wolfe Action, 1.250" straight taper barrel 34" long with a brake, barrel blocked, McMillan 50 cal light gun stock and a Leupold 8.5-25 x 50. It weighs 23lbs. If your looking to build a rifle around this weight I imagine your not far from the vehicle when shooting. Why not just build it to what you'd like for "bench work" and be done. For the bigger calibers the heavier weight will help reduce the felt recoil. For .30 caliber the .30-.378 Weatherby, .300 Lapua, .300 Ultra Mag, .30-.338 Winchester, or .300 Weatherby with 240 SMK's will get you out there with plenty of energy to the ranges you'd like. If you want to still reach out there and then some jump up to .338 cal. The .338-.300 Ultra Mag (aka .338 Edge), .338-.378 Weatherby, .338-.416 Rigby, .338-.408 Cheytec all are great choices for ultra long range work.
The .300 Lapua AI shortened you mentioned would work, but why go through the headaches of more brass prep work if you don't need to. If you like the .300 Lapua AI I'd reccomend just shoot that. Load up the .300 Lapua and blow out the shoulder.
What are you hunting at this range?