A .17hmr should do fine up to 150 yards, but you are not giving up a ton of yardage with the 22 mag rf.
I understand some 22 mags can take a barrel swap for the 17. You might want to explore that avenue if money is tight (when is it not?). I don't know if you can swap with the Savage rimfire, but I understand you can with their centerfire offerings.
The .223 would almost double the range of your 22 mag at the expense of more muzzleblast and more expensive ammo. It would give you a good coyote gun too. That's the route I would take if I were you. But I'm not and the 17hmr will do what you need at less expense.
Best of luck
la plus belle des ruses du diable est de vous persuader qu'il n'existe pas!