Tough Hogs, Tough Bullets
I have zero experience with Sierra GameKings, so I can't offer anything on those, but I will say, I'm a Norma Oryx fan when it comes to lead core/copper jacketed bullets. Heavy, Tapered Jacket and a bonded core. I shoot .270 Win. exclusively in bottleneck rifle, and the 150gr. Oryx is incredible... Deadly on tough pigs.
As Your 30/06 is intended to be used as a BrushGun, try the more expensive Oryx, as they're offered in several weights for .308", BUT if Your wanting to keep cost down, I'd use Hornady Round Nose Interlock in Your prefered weight. I think they're offered in more than 1 weight for .308".
In any case other than the Hornady RN, I would use nothing less than Bonded Core (for leaded bullets), huntin' tough squealers...
I have a Browning A-Bolt Stainless Stalker that recently received a Douglas Air-Gauge #4. My load for .270 Win. with 150 gr. Norma Oryx... (if You have/load for .270 Win.)
*53.0 gr. AA3100...seated to 3.300" C.O.A.L. (may contact lands with a rifle that has very little freebore)
.75" to .95" at 100yd.s. .75" if I haven't had too much coffee...
Wish I could give more advice for .308".