I checked Sierra's website and I see a 350g Matchking, but no 350g Gameking. Are you referring to the Matchking?
I know Berger is planning to introduce a new .375 bullet, but I don't know anything about the planned weight or bc.
I ordered some 300g Cutting Edge bullets for my .375. They have a similar bc to the Sierra Matchkings. I believe I should be able to push them to around 2700 or so from my H&H. The down side is that they will be expensive to shoot. They will definitely not be plinking bullets!
I have been thinking about the long range potential of my H&H and plan to experiment a bit to see what it can do. If the CE bullets work out, I hope to use Bergers or Sierras as a less expensive alternative for practice bullets.
Of course, if the CE bullets work out, I will also have to switch to long range glass for my .375. Right now, it is set up as a short to medium range smasher.