re: Deer Hunting Rifle $2,500 Budget

Montana Rifle Company (Out the Box)
Winchester Super Grade - and you could send off for blue printing accurizing.
Zeiss Conquest, Vortex, Better grade Burris Glass
That will get you close to that mark.

You could acquire the action of your choice and send to Pac-Nor or someone like that, wait 6 months at best, and build one yourself.

As far as the Sako Remington suggestions they are solid but I personally am not much of a push feed fan.