both will work
Both the 7.62x39 and the .308 will work. It doesn't matter what caliber is used, you cannot absolutely guarantee that the animal will simply collapse on the spot. This is why tracking skills are just as important as shooting skills.
The .308 is probably more "versatile" overall.
Whichever you choose, make sure it is sighted in;
- know where the vitals are on the animal.
- If it does run after the shot, don't panic. Memorize where it was when you shot, and where you last saw it. Be patient and wait a bit. If you check and find blood, mark it with toilet paper. You can make a visible trail using the TP which will help guide you. Don't be surprised if the animal goes over 100+ yards with a shot through the boiler room.