Based on my 338-378 you may want to check the nominal amount of factory bedding. If the rifle has been shot much you'll probably see that the daub of epoxy they use has cracked and is useless. You're better off to fix that now than chase it down later.
At 200-600 yards there are a wide variety of possibilities. A 4.5-14x50 Zeiss comes to mind if you buy one with target turrets. Doing that you can spend $600, or work your way through various scopes up to almost $2000 for a 5.5-22x56 Nightforce. Your budget is the only thing stopping you.
Your current Alphamount is excellent. I'd call Near and see if they can tell you if it accommodates an objective larger than 50mm.