The 168gn Berger Hybrid is a very capable long-range bullet with an excellent BC. You'll get, or should get, over 2,900 fps in a 30-inch barrel using VarGet. The Hybrids are not as easy to tune as the Target BTs in my experience, some users say otherwise. I found that I had to treat the 168 Hybrid like a VLD and seat it into the lands to group well in my rifle.
An alternative to this bullet and to the 155s is the 175gn Berger LR BT that is usually easy to tune and although not as ballistically efficient as the Hybrid numbers is still an excellent design. Recoil remains moderate with both 168 and 175gn models, many sling shooters using the latter for this reason in any bullet rules US Fullbore and Palma Rifle long-range matches.
The 185gn Berger LR BT 'Juggernaut' is, like its smaller 155.5gn sibling an outstanding once in a generation design that really works well at 1,000 yards especially if you get MVs into the 2,775 to 2,850 fps window. You do start to notice significantly heavier recoil with it though. It's also such a popular bullet it's regularly out of stock with Berger struggling to keep up with demand.
As you have 'shoulder issues', I'd stick to 155s and look to get them shooting small groups and tight elevations at 3,000 fps + MV. Your rifling twist rate isn't a real issue such is the quality of these bullets now. The 155.5gn Berger usually provides the best mix of precision and easy 'tunability', but others from Lapua and Sierra are strong contenders. The 155gn Hornady A-Max is best avoided for long-range precision shooting as it has a short, too steeply angled boat-tail section that causes transonic speed instability.
I'll be honest though and say you WILL struggle beyond 800 yards with a 24-inch barrel rifle no matter what you use. 24 inches in .308 Win just won't provide enough velocity in any bullet to perform really well at 1,000 yards. You will be able to keep the 168 Hybrid and 175 LR BT supersonic at this distance with top end loads, the 155s less likely to do so. However, transonic terminal speeds often see bullets more heavily affected by wind changes, groups open out, and crucially elevations drastically increased. 24-inch barrel rifles with good handloads tend to still work well at 800 yards, but struggle increasingly the further the target lies beyond that distance. That's not to say you won't hit the target, and you can still have a lot of fun. (The other factor too is the elevation of the ranges that you'll shoot over and likely air temperatures. Raton at nearly 7,000 ft ASL changes 308 ballistics dramatically for the better because the thin air reduces drag on the bullet.)