I dont know how much you have looked into these bullets, but depending on the gun, they will either shoot incredibly or terribly. My 338 shoots well under a MOA at 100 yards with the Barnes TSX, but my 30-06 shoots about a 6 inch group at 100 yards with them!!! Do some experimenting, your gun will either love or hate them. I think Barnes are great bullets and wish my 30-06 would shoot them, but no luck.
Another thing to look at as well, copper is lighter than lead, so the overall length of the bullet is longer than the same weight in a lead core bullet. The longer the bullet the harder they are to stabilize in flight. So if you try a heavy for caliber bullet and it does not shoot well. Get a lighter bullet and try that, as it will be easier to stabilize.
Barnes claims that you can drop a bullet size (i.e. a 165 grain instead of a 180 grain, in a 30 cal) because they retain 95%+ weight instead of 60 or 70 percent in a normal lead core bullet. The thought being that the retained weight will bve the same.