I haven't shot elk with the 300 Win and 208 Amax's so I get that out up front. But I do have quite a bit of experience with Amax's that may be helpful. You shoot them too so this may be stuff you already know but just in case.
Cup and core bullets like the Amax aren't for guys that like to shoot at any animal they see at any angle. They harken back to the good 'ol days of hunting when guys actually took the time to wait for a good shot before firing away at an animal.
If there is a negative side to all the premium bullets out today I would say that is it, guys end up taking marginal shots on animals hoping the bullet will make up for the other shortcomings they are introducing to the shot.
So, if you wait for a good shot, and stay away from the point of the shoulder you will find much, much more success with Amax's, even on elk. Slightly quartering to or away are fine but I don't go much past that because with severely quartering shots you introduce too much rib bone and tough hide at bad angles and increase the distance the bullet must travel to get to the vitals. All of that simply reduces your chance for a clean kill with any bullet let alone with frangible ones.
Talking hunting bullets is an interesting thing. Guys run to a specific brand of bullets and swear they have been created with magic pixie dust when in fact they are very simple tools. Some of which have been around for a very long time. Berger is a good example. They no doubt have some specific design characteristics but for the most part they are an old style cup and core bullet which is the kind of bullet that has been used for decades. They choose that kind of bullet because they are the most accurate and they found out, quite by accident I think, that their bullets works very well on game when used correctly. Interestingly, the Hornady BTHP performs very similar to Bergers on game mostly due to the meplat design and how it delays expansion slightly before blowing up like a grenade. Me personally, I like that kind of bullet. For the game I hunt in North America I don't need a solid bullet getting pass through penetration at extreme angles. I want to anchor them where they stand if I can. So the Amax's will do that. The plastic tip on the front opens the bullet up faster than the HP's so you have to keep that in mind. Heavy for caliber bullets like the 208 Amax are the way to go for elk IMO as the extra weight ensures as much penetration as possible with that bullet. I have seen elk taken with lighter weight Amax's but the safe bet in my opinion is the heavier stuff. You have the cartridge to shoot them so no big deal for you anyway.
The biggest issue you will face is close up shots. There are far more examples of Amax failure up close than at medium to LR distances. Again, the extra weight of the 208 will help with this. Just a thought, the Hornady BTSP's or Interbonds shoot very close to same POI of the Amax's in most guns out to 300 yards or so. I often times carry two loads for this exact reason.
At LR is where the Amax bullets shine IMO. They are every bit as accurate as any bullet out there in my experience. And you can expect good expansion at velocities well below other bullets at LR, much lower than what Berger states at 1800 fps. Many guys want a bit more velocity than that to be safe.
I have killed and seen a ton of deer taken with Amax's and quite a few elk too. But again, good shots were taken which is a big deal to success in this guys opinion.
Good luck on your hunt and your decision on bullet choice.