If brown & polar bear are on the list, 338 or 300 WIN MAG would be my choice of chamberings. Bison and moose are big, but they don't tend to chew on you if you screw up, so for a do-it-all I'd likely pick a 270 WIN or 30-06.
Rifle limited to $1k means either quality old and used (Brno ZG 47, Win pre-64, Husqvarna 1600/1640 or Mauser 98) or new manufactured (Ruger M77 MKII or Howa 1500/Weatherby Vanguard). I just can't seem to find many quality bolt rifles under $1k newly manufactured that don't have one or more features that I don't like.
