I had a Vouree Mauser in 243 some years back. It had a thin barrell with a diameter of my little finger. With scope, a bushnel banner. it weighed 5.5 pounds. It would place the first three rounds in a 3/4 inch group at 50 yards.
After that they opened up to about 2 inches as the barrel heated up.
For hunting this is good accuracy. You might check with some of the Barrel makers and get the thinnest barrel they make.
If you have a wood stock you can drill holes in it and remove a fair amount of wood without destroying the integrity of the stock.