No one has mentioned yet (that I have seen) that if you go with the .338 Edge and shoot the long high bc bullets like the 300 Berger, you will likely be confined to a single shot. A 4.00" Wyatts mag box is an option, but many people seem to be unhappy with them, and in some cases you may want to even load longer than 4" OAL. The .338 RUM will get you within 100-150 fps of the EDGE, maybe closer, and can run as a repeater with the Wyatts 3.850" mag box. If you run the ballistics, a 300 berger at 2700 vs 2850 is not that big a difference. Just a thought, if you don't want a single shot. If you are fine with a single shot, I would go with the EDGE. Another option is as follows, you can find all these components in this price range:
Used .338 type action (stiller TAC 338, lone peak Razor .338, etc.) - $800-$1000
Custom steel barrel from Krieger, Brux, or any of the other manufacturers - $325-$400
Stock from Boyds, or decent used composite stock- $135-$200
Trigger- $135-$200
Bottom metal or DBM - $200-$300
Brake- $75-$235
Smith fees to assemble -$300-$500
That is $1970-$2835, so you could make it come in at your budget, less if you already have a rem 700 bottom metal or any other donor parts, but for a full custom that you could run the .338 Norma in, send the same 300 bullet at 2700-2800, and easily run in a long action with essentially unlimited seating depth, and as a repeater. Cost will be the same as a rum or edge, brass is pretty much the same price, same bullets, pretty much same amount of powder, and same primers. Use peterson brass, you should get around 10 loads or more per brass if you load reasonably. And the Norma is notorious for being stupid easy to get shooting fantastic. You could however, do the same and get a basic used Remington Long action and run the edge for probably $500-$600 less all the way around. Just some thoughts.