Just go and buy a complete Handi in .308. Then you don't have to mess with changing barrels.
The .308 is a good round, I carried one for 14 years, it was the only rifle I owned back then. 165gr bullets are the optimum in my opinion, for big game. 125 and 110gr bullets work great for varmits. 180gr is OK if you are close. 200 and 220gr are just too heavy. I took Mulies, Whitetail, Antalope, Coyotes, Moose and Caribou, with it. Then I got an 06. .308 hasn't been shot since, it's been a vault queen now for 27 years. Now that I think about it, think I'll clean it up and give it to my oldest son. Time to start downsizing.