Simply put, no, you won't be happy with an AR-10 that converts down to 5.56. It has been done, and can be done, but you will spend far far more on that set up than you would have spent on an AR-15 and an AR-10. It's just not worth it to hybridize an AR-15/10
That said, there are certain rounds available for the AR-15 that you would be happy with for big game hunting.. .243wssm, .25wssm, 6.5 Grendel, .458socom, 6.8spc, etc
I'm not sure I'd be interested in a 7.62x39mm AR-15, there are better cartridges available.