To go from 06 to 338 Win isn't as simple as just replacing the barrel. You need to change the bolt too since the base dimensions of the 2 cartridges are different. I was going to turn an old 7mm Mag into an 06 and the gunsmiths all told me the bolt needs replaced too. he 338 Win base is the same size as the 7mm Mag base so you would end up with the same problem.

I'd suggest going to the 9.3x62 Mauser instead. The base of both the 06 and the 9.2 are the same and the 9.3 is a great round that has been around for well over 100 years and is very, very popular in Europe and Africa. You can get heavier bullets in the 9.3 than you can the 338.