As a rule, I think you should target shoot with the bullets you intend to hunt with. You did not say what game you intend to hunt, but the .308 is a capable whitetail/black bear round.
The Nosler Ballistic Tips are often said to often be as accurate as match bullets, and my experience with them tends to support this. These bullets in 165 gr weight would do very well on medium sized game, and will produce small groups if you do your part.
If moose are on the menu, I think you need to use the best bullet you can find for your .308. While I like the TSX, even the lighter ones tend to be quite long. Due to this length, the TSX's might have to be seated deeply, taking up valuable powder capacity, in order to function in your magazine. If this turns out to be the case, a 180 gr round nose might be a better choice for moose. A lead core, round nosed 180 gr bullet does not need a bonded core to produce reliable terminal performance when loaded to .308 velocities. The round nose flat base bullet is the shortest design in any given weight, and the simplest to design for reliable performance at moderate velocities. You will need to shoot these enough to learn the trajectory, but seldom are moose a long range problem, and a 150 yard zero will work well. Remington Corelokt or Hornady Interlocks are both good choices in round nose bullets.