I think a standard .280Rem./Exp. would work just fine = better than 7x57. How about a .280 Flanged Nitro Express same ballistics as a warm loaded 7x57. 160gr. @ 2,650fps. - page 295 in my newer, 6th Edition COTW

.260 makes a nicer bolt gun, I think.

The longer case, with long slim bullet & rimmed case - nice!

If going shorter, full stocked Mannlicher, then the .260 Rem. for sure.