Do something fun, and for a little less recoil than the 340 Weatherby you can get similar ballistics with the 358 Norma.  I built one a few years ago that would lauch 250 grain bullets at 2800 fps.  That's not fooling around.  And while stout, it was pretty manageable.

My big question would be are you in a position to where you could use a 300 or 340 Weatherby more than a couple of times?  That along with using them for long range hunting you will need to do a lot of practice with either round to be capable of taking a long shot on an elk.  That and the ammo is getting more and more expensive for both rounds unless you reload.  For the .340 you are looking at $90.00 for a box of 20 for premium bullets and the .300 isn't far behind. 

As for me the .340 Weatherby is the best all around factory cartridge for hunting elk that I have found.  It will shoot as flat as a 7mm Remington mag with 250 grain bullets with a lot more punch in the end.  But the big thing you need to look at is the availability of ammo for any round you decide to get if you don't reload. 

But if it was me I would get the .340. 

I don't have any experience with the big calibers so I won't try to give a recommendation there but I did want to compliment you on the rifle. I had heard of Shaw barrells but didn't know they were building complete rifles now. I looked up the website and it looks like some really nice rifles witha lot of different options available.

How long is it supposed to take for your build to be done?  It may be shorted because of the Nra auction but I have seen waits posted from 3 months to over a year. It may be something I would be interested in as well but not if I had to wait so long after placing the order.