Getting to 1500yds really isn't that hard anymore with todays components. We have done it with out 6mm's and even stretched it out a little farther than that with them. Because of your requirement of taking game at those kinds of ranges, that really does put you into 338 Edge territory. I say 338 Edge because it outperforms the Lapua and has cheaper components to do so.
Now with that being said, I don't know your skill level, but I can tell you it takes a ton of practice to be effective at 1500yds. This takes shooting. As Carl said, taking one of Shawn's classes is a great start, but even he will tell you that you need to crawl before you walk. My suggestion would be to get a Remington 700 action, and build either a 7RM or 300 RUM. Shoot it until the barrel goes out, then you will have the confidence and skill make the 1500yd+ shots on critters. Not saying the 338 won't do the job at closer ranges, but its way overkill, especially for 200lb whitetail. The 7RM with a 180 Berger actually shoots the same dope as a 300RUM with a 210 at 1K. It has slightly less energy, but the same drop and drift with way less recoil. Our new 7 Rogue with a 180 shoots flatter and less wind than both of them with still less recoil than the RUM. The Edges are the #2 build I do for guys, and surpisingly enough, I have bought a few back because they can't shoot them. To much recoil and blast for them. They end up with a 7mm at that point and are happy as hell. As much as we like the Edge for our style of hunting, we have started using our 7mm's more and more now for the area's we know are going to be under 1k. The Edge comes out when conditions are poor and the ranges are going to stretch out much farther.
Lots of choices out there. Pick one and shoot the hell out of it.