I used to hunt in Fredricksburg around 1955. A .243 is enough gun to take your local deer, but your .308 is probably a better choice.
There's more than one way to reduce felt recoil without buying a new gun/barrel.
As mentioned, you can load the butt up with mercury vials or lead to create more weight for the gun.
You can shoot lighter bullets, like in the managed recoil loads, but the BC get's less than optimum for longer shots (that shouldn't be a big concern where you are).
You can shoot the same bullet with less powder, but you have to reload to take advantage of that.
You can replace the recoil pad with a Limbsaver pad.
If you don't reload, more weight, with managed recoil loads and a better recoil pad should make things more comfortable.
Be sure to wear ear muffs or ear plugs, since muzzle blast probably accounts for more flinch production than actual recoil.