Any 165 grain hunting bullet will work out of either cartridge. It's a Ford vs Chevy thing, so what brand doesn't matter much. You load for the weight, not who made it, its shape or construction. No beastie will know or care what bullet put him in the freezer anyway. Plus velocity isn't as important as accuracy.
Ballistically the cartridges are identical(the .308 was designed to replace the .30-06 but required the same ballistics.) with like bullet weights. They both just happen to like 165's. Difference is about 100 fps. Not enough to worry about. A 165 will kill any game you care to hunt, bug bears included(would prefer a 180 for them though). Never bothered with premium hunting bullets myself. They don't do anything any other bullet won't and see no need to pay extra for a brand name.
Partial to IMR4064(use it with match grade 168's out of my 1903A4. Sniper's rifle from W.W. II.), but it's not the only game in town. There are hordes of powders that work well with either cartridge. Look in your manual for the one given for the accuracy load for a particular bullet weight. Mind you, not all manuals have that.
My vote is for whatever shoots best from your rifle. A lot of bullet performance mumbo-jumbo, IMHO, is baloney. If you aren't confident with the loads accuracy don't shoot. Place the shot and even FMJ will devastate an animal. I personally use Hornady InterLoks. Inexpensive and seem to always give good groups. I shoot .35Rem, .308W and 8X57.
As far as "lead-free" goes...P.T. Barnum had it right...
Both of these posts heavily weigh the importance of accuracy as the most important feature of game bullet performance. IMHO, shooting at live targets, under typical field conditions, places far more weight on the importance of bullet design, construction, and terminal performance than it does on mechanical accuracy. If a rifleman practices incessantly with a good rifle, and manages to maintain an average of 2 MOA under field conditions, which by the way is very good, any potential mechanical accuracy under 2 MOA then is irrelevant to his problem. He can't make use of it. A 4 minute rifleman armed with a 1 MOA rifle is still a 4 minute rifleman. Few modern game bullets shoot worse than 2 MOA with any reasonable load in a good rifle, provided the bore's rate of twist is fast enough to stabilize the bullet, but there are plenty of super accurate bullets that produce miserable performance on game.
What the hunter can make use of is a bullet whose design and construction is well balanced for the velocity of his rifle, the range at which he'll shoot, and the density of his target. Clearly a bullet that impacts at 2000 fps doesn't need to be constructed as strongly as the one that impacts at 3000 fps. Reasonably the bullet that explodes the heart and lungs of an antelope isn't the best choice for a bison, although admittedly these represent the extremes of the North American hunting experience. A .30 caliber Hornady Interlock bullet in the 150-180 gr range is suitable, and in most cases an excellent choice, particularly at impact velocities typical from .308 and .30/06 rifles, but there are exceptions. One exception might be where the cartridge is a large capacity case and the bullet impact velocity exceeds 3000 fps, and where the target weight approaches 1000 pounds. A Barnes TSX or TTSX can weigh as little as 130 grs, and have an impact velocity in excess of 3500 fps, with no loss of performance, where the cup and core bullets, even good ones like the ILs, will shed their cores and produce nasty shallow wounds which have little chance of anchoring a large game animal. Another advantage the TSX has over a lead core bullet, is that it will expand at lower velocity, where the lead core bullet usually needs a minimum of 1700 fps, and arguably 2000 fps to reliably upset. But more importantly, choose your game bullet for the performance level of your rifle, and the density of the game you intend to hunt.
As to the assertion that FMJ bullets devastate game, well it depends. First there are non-expanding game bullets, hard cast, jacketed, and mono-metal designs which are as effective as expanding bullets, particularly when the cartridge in question is marginal for the game hunted. These bullets are typically short for caliber, have parallel sides, and are blunt nosed. They penetrate deeply in a straight line, even when bone is hit, and produce large wide wound channels until their velocity decays to 1000 fps or less. If the reference is to FMJ military bullets, it depends on how the bullet in question performs. A pointed bullet that pencils through, has little immediate effect on the game, even though the shot might prove fatal over a period of time. The long FMJ spitzer which is base heavy does have a tendency to swap ends upon contact with a denser than air medium. This extreme yaw will undoubtedly result in massive soft tissue damage, but should a major bone be hit when the bullet is side on, penetration might well stop then and there. Undoubtedly this non-uniform performance is the reason why the majority of jurisdictions have banned the use of FMJ military bullets for use on big game.