I feel that you may need to get a load worked up for the rifle that is reasonably accurate and do some testing with a few different bullets and see which is going to be the most accurate.
The problem you face with a fast 30 such as the 300mag is complete devastation of animals at close range if you chose the wrong bullet. And remember, there are 5 animals taken under 100yds for every one taken at 500. That means you need a bullet that will hold together through a high velocity short range hit and not put a hole in the animal the size of a softball, yet still perform when the yardage starts to stretch. Bonded and sectioned bullets will do this far better than a traditional bullet, as will most solid copper or gilding metal bullets. The Berger may do well in this area as well, though I have yet to really test it much myself. This leaves a few distinct choices that are proven to be very effective, Nosler Partition, Nosler Accu-Bond, Barnes TTSX and TSX, Swift A-Frame, and possibly the Nosler E-Tip, Hornady GMX or Hornady Interbond.
I don't feel Sierra is truly up with the times in the hunting bullet arena, though the GameKing and ProHunter are good designs, there are others that are a bit more advanced than the cup and core design of the Sierra's. Just my opinion, so please everyone refrain from flipping out on me, as it will not change my mind. They are more concerned with making the SMK shoot bugholes and producing a good varmint bullet than designing a hunting bullet, which is how I see things.
That leaves you with several very solid choices to find a bullet that will shoot accurately from your rifle. I personally have yet to see a rifle that won't shoot a Nosler Accu-Bond accurately, and the Hornady Interbond is similar in design, so starting there isn't a bad idea. The A-Frame and Partition are of the same basic design, so whichever is more accurate of the two is what I would take if you go that route. The Berger has to fit in somewhere, so here it is. Then you are left with the solid designs, the TTSX, TSX, E-Tip and GMX. I have little experience with these and understand the solid design is very tough and performs well on game, but can require a bit of extra pressure to hit the same velocity as a comparable weight bullet.
Every bullet I have listed has taken the game you desire, and some like the Partition and A-Frame have been doing it for decades. You will not go wrong with any of the above.