I don't like the 5.56, but that's my personal opinion. I really do like the 7.62X39 and like the 7.62X51 even better. 7.62X39 is cheaper, but with the cost of good AK's today, if I were to go shopping for a shtf rifle I'd probably go with the Springfield Armory Scout rifle with 18" barrel.
At one time I was tempted to dump everything I had in favor of a one-rifle-do-it-all rifle in a XCR in 6.8spc. I like the 6.8spc, and would've given up .308, .30-06 and 7.62X39 just to have that one caliber as I felt it could do everything I'd ever need a rifle to do. But I came to realize that the 6.8spc had not gained enough popularity in the civilian ammo market to be able to acquire easily or cheaply.
So of the two calibers you chose, I'd go with the 7.62X39 over the 6.8spc because of price and availability. But if I were you I'd also look at the .308 which clearly outperforms any other caliber mentioned here.