Really, one bullet for coyotes that weigh around 30 lbs, deer at 150 lbs and Elk at 500 to 700 lbs.? You want to shoot all of them at ranges up to 400 yds.! Assuming you have a good barrel and re crowned the muzzle properly the bullet weight selection with Elk on your list pretty much makes a heavier 170 to 200gr bullet the choice.
You can buy Hornady 125gr Spitzers for varmints and push 'em to 3,000 fps for flat shooting long distance but deer needs a bit heavier and elk is a large animal and is better shot with a deeper penetrating heavy bullet.
Personally I'd forget about 1 bullet and use 125's for the Coyotes which is likely the only thing you will hun 11 mongth out of the year and put together a heavier bullet load for the deer and elk season and resight the scope in. Just record your changes in settings and change them back to the lighter bullet after season.
Heavy bullets driven hard kick like a mule and are more expensive than lighter weight for varmints.
As for powders I've used W748, IMR4064, and AA223.