If you have $4k for a budget, drop the Rem 700, order in a Borden Timberline Magnum receiver, McMillan stock and any one of many top end barrels.
In the end, the Borden is a world class barrel, especially for the RUM or Edge class of chamberings and not much more then a Rem 700 once you get a Rem fully trued and ready to build on. I suspect if your wanting to spend up to $4K for the rifle, you will have the Rem 700 fully tuned and accurized for the build. Your money would be better spent with a full custom Borden receiver, hands down.
As for the chambering, either will get well past 1200 yards. Both are very accurate, the Edge will get you another 100 fps over the standard RUM. In a 28" barrel length, the Edge will get you around 2800 fps, the RUM around 2700 fps.
The Edge is worth it as its very easy to load for, just as easy as the 338 RUM but a boost in performance.
I will say this also, the 338 RUM brass, being the odd ball of the RUM family will likely go the way of the DOO-DOO bird in not to many years so if you plan on using this rifle for many years, an Edge will be a better choice as there will always be 300 RUM brass available to us.
Again, if your going to invest upto $4K, do it right and get the best bang for your buck, go with a custom receiver and you will never be unhappy with your decision.