Try this one on. Since you have $1300 budget, you can get yourself a custom rifle
Get a Savage in a RUM. Spin on a Pac Nor rechambered barrel. Even swap in a Boyds stock. You should still be under your budget.
however, with a bedding job and some load work up, most Savage or Stevens will shoot very well. Now you can have a rifle and scope for $1300 or less.
If you don't need a light trigger, a Stevens 200 in 300WM will do dual duty. Just not a lot of shots before the barrel heats up. But then, you can always swap barrels.
If you decide that a single shot is ok, wow, we can really drop your costs and up your options.