Well, as a sheep guide (and even been on a few Mt goat hunts), I can tell you that the 300 rum is WAY overkill for either bighorn or the thinhorn sheep at the distances you indicated. And it's a complete waste of powder in a 22" barrel. And, as others have said, it's brutal in a lightweight rifle-even with a muzzle brake (you didn't mention if brakes were an option and I don't think anybody asked you that yet).
Contrary to what some might believe, goats are not thick skinned nor are they very big. Most of what you see if fur. They are stocky and thick through the chest, but regular hunting bullets from non-magnum guns kill them all the time, no problems.
I have let friends and clients use my rifles to kill sheep at various distances with various bullets. Of them all, I have found Berger VLD's to be the best overall bullet for sheep. Accubonds are a close second-especially if distances are on the shorter side.
I have to disagree with Alinwa (yet one more time) that the 270WSM is a ultra fast round that is hard on bullets, and I also disagree that the only bullet that can hold up to it is the Barnes TTSX. Accubonds are bonded bullets that generally have higher bc's and don't come apart at any speed. If you absolutely must have an all copper bullet, Nosler also does a bullet called an E-tip that is almost identical to the TTSX except for the grooves. And, of all the WSM's, the 7mm WSM is ballistically superior unless you want to shoot bullets under 140 grains. Even at that, the 7 can use bullets in a much wider weight class which also makes it more versatile.
Velocity does not mean energy, bullet weight means energy. But you don't need as much of it as you might think for the distances you are intending to shoot.
The 7mm-08 is a sheep cartridge if there ever was one. You can't go wrong with it in a 20" to 22" barrel. Another one to seriously look at is the 6.5x47 Lapua shooting 130 or 140 grain Berger VLD's. Deadly accurate, plenty of power for sheep out to 500 yards, less recoil than the 7mm-08, and higher BC's in bullets that weigh 140 or less.