Proof of a hunter safety course is not required to buy a license. But, for anyone (with the execption of those hunting on their own property) born after Jan. 1, 1969 are required to have proof of passing a hunter safety class before hunting. His MO class will work for that, but he must have proof of that class on his person while hunting.
Once he has been a bonifide resident of TN for 90 days, he can purchase a TN license. Since he is required to give up his MO drivers license and get a TN license no later than 30 days after moving into the state, he will be able to use that after the 90 day mark to buy his TN license.
