The 208 Amax is just about worthless for hunting...it won't expand at subsonic speeds, so at best you are stabbing the animal with a .30 caliber ice pick. Yes, there are people who have taken animals with it, but it is very reliant on perfect shot placement into something vital like the heart or brain. If you really want to hunt subsonic look at some of the expanding subs out there now. They are a bit pricey, but worth it to ensure a good humane kill.
It does work well for subsonic plinking however, but keep in mind that it will have a ton of drop at 200yards...something close to 5 feet or more! You'll basically be lobbing artillery which could be fun!
I hunt deer and other critters with the 300BLK and have personally decided to go exclusively supersonic for hunting for a couple of reasons. I've harvest 5 deer with the Nosler 125gr BT bullet loaded at about 2100FPS.
1) More practical trajectory inside of 200yards (with a 50 yard zero it will be ~6" low at 200), with my own limit being 150yards for bullet performance
2) Energy = 1/2 * mass * (velocity^2). I'm loading supersonic bullets that are 1/2 the mass but roughly 2x the velocity of subs, meaning double the energy on target
3) At super sonic speeds there are bullets that expand and perform every well
With a can you will have no problem still getting follow up shots. The super sonic crack often scares things initially but often times animals like deer aren't sure what the sound was and may come right back to the spot 5 minutes later.
Powder selection is the key...with heavier subs the powders like 1680 and H110 work fine, and those also work great for supers as well.