Hello Wrought,
I am just a bit North of Wyosmith, and there is a LOT of big game shot here in Big Sky country as well. In your price range I second Wyosmith's choice of the Ruger M-77 mkII. I have less experience with the Winchester, but I think they cost quite a bit more.
I have nothing against a fast .25, and indeed if you are going to shoot sod poodles and woodchucks with your deer rifle, the .25-06 is a great choice. If you'd like to hunt elk or moose with the same rifle, then the .270, .280, and .30-06 start looking much better. The .25-06 will kill elk, it's just that the others are...well a bit more appropriate for heavy boned game. Moose, while larger, are usually not regarded as being as tough as elk. The go-to factory load if you must shoot elk with the .25-06 is the 120gr Nosler Partition. Plenty effective on deer, too.
At 300 yards there is nothing wrong with any of the above cartridges and my .30-06 has felled many deer and antelope at that distance, this with a 200 yard zero and a back bone hold.
Good luck, and let us know what you pick.