Honestly any of those would work. However, none of them should really be considered unless you are going to shoot 1200 plus. Energy and wind drift are where they really shine. I wouldn't do anything until you shot your 6.5x47 for a bit as it will be a WORLD of difference compared to your .308 and may very well do everything that your expectations of this Magnum rifle will achieve.
However, if you want to proceed beyond that. Of the 300's the Norma is going to have more horsepower than a 300WM and will get the same bullet going faster. However it is on a Lapua Magnum Bolt Face so some actions are out of the question. Of those 2 I'd still choose 300 Norma.
For the 338's it is a bit of a toss up. The case of the Norma is slightly shorter which has an advantage and a disadvantage. The pro is that a 300 grain bullet will be able to be seated properly and still be mag fed out of Non-CIP length magazines. However the Lapua has more case capacity and hence it can get a 300 grain bullet going faster which equals less drop, less wind and more energy, albeit probably somewhat negligible in the grand scheme. With Lapua announcing they will make Norma Brass for the 300 Norma as well as the 338 Norma there is no advantage there. However more components are "around" for the 338 Lapua being it is a NATO caliber. And if you ever wanted to sell it I'd say the Lapua would have a broader audience. If you do build I would go with CIP length magazines and an action that accommodates them as it just give you more options for bullet selection an OAL of rounds.
You also could buy 1 action if you went 300 Norma and get either a .338 Lapua or .338 Norma chambered and all you would have to do is a barrel swap because the both take the same magazines as well as bolts. Very easy to do with a barrel vise such as a (Viper Barrel Vise $58) and an action wrench $80.
I would go with a Surgeon XL action but I am bias. For what it's worth I have a 300 Norma and a 338 Lapua is being built as we speak. Both on the Surgeon XL Action.
http://shop.otmtactical.com/VIPER-BARREL-VISE_p_360.html
http://www.bugholes.com/product-p/338-bug-aw.htm