I'd go 6-06 or 25-06. My last (and only) rifle built with goats in mind was a 25-06AI of the Faux-Ti flavor with a PN Mt Rifle contour at 23".
I ran the 100gr NBT's in it and it was wicked on a couple goats. My wife shot one prone at 416yds and he simply crumpled.
I shot one at 460yds sitting with shooting sticks and he too crumpled at the shot.
I shot a couple whitetail bucks last fall with that rifle as well and it did what one would expect.
I also learned that in all reality the 25-06AI w/100's really didn't do anything (for me) that my 243/243AI's couldn't do with 105's. So I sold the rifle and still am running my 243/243AI's.
I guess if I was doing it over, and on a LA, I would go 6-06 to take advantage of the high BC 6mm bullets. Good luck!!!