That makes it pretty slim pickins.
Unless you can find a great used rifle that has glass already, you're looking at the entry level rifles such as Savage, Marlin, and Ruger. Nothing wrong with them. They harvest lot's of critters every year.
If it were me, I'd look for a good used Remington 700, or Ruger M77. There are others out there as well. A friend recently found an old FN Mauser action JC Higgins at a pawn shop for $300.
He does not hand load, and has only tried factory loads. Some were sub moa, most were right at 1" or 1.25". Pretty good shooter for $300 including the scope.
This is just my opinion, and there are MANY of them here on the fire. Good luck to you and your friend.
