It would be simple to do the .338-06 A-Square on that strong action. Just a barrel without fooling with the bolt. I would imagine that you would have feeding issues if you tried to use a .30-06 and a .338 WM in a switch barrel outfit.

I have never had a .338-06 but have toyed with the idea. I use a .338 WM quite a bit and the idea of having a lighter rifle with less recoil, and only giving up 200 fps or so sounds attractive.