The .300 is a better choice than the 7MM. Just take it to the range and practice at 200 and 300 yards. When you are out on the Tundra, the Caribou will see you first most likely. If they have been shot at they will run the other way. If no one has been shooting at them, they are curious animals, and are likely to approach you. If they do that just sit down, everyonce in a while wave an arm, ot something to keep their attention. The biggest bulls will hang back, but the young bulls and cows will often approach to within 40 to 30 yards. You may have to take a 200 or 300 yard shot at the big bulls hanging back.
I haven't had much experience with the .300, but I often use the 165gr boattail for Caribou with my 30-06. Yes, Caribou are not hard to kill. If hit solid they usually lie down, and if not disturbed will die right where they lie. You just need to be aware, once you start skinning, blood will carry far on the wind. Keep your eyes open for unwanted guest, of the big brown kind. Odds are you won't see one, but better to be watchful and not see one, than not pay any attention and get visited.