I am a certified 33 loonietic, and so have worked with the 338/06 (not the AI) and the 338 WSM and the 338 (as well as a couple of bigger ones).
Personally, for me the short action thingy doesn't do a thing for me. The smallish amount of weight saved and or the bolt throw being a bit shorter doesn't mean anything to me.
The 338/06 feeds like a dream and mine with a 23" tube would do with a 225 about what a 338 would do with the 250. The WSM closes the gap even more.
My WSM feeds well, but having 3 down is a tish bit tight but do able. (M70, Schneider tube cut to 21")
I do like the fact that the 338/06 can handle more rounds down as I don't shoot very well....grin
After years of using the .340 I found the 338/06 to be a welcome and wonderful change. Way more user friendly.
As for shooting at range, any of the 3 can be made to work to 500 yds with no problemo with dotz and or the Burris BP. In my 338/06 I ran the BP and had no problem in sighting it to kill elk to 500 with a 250 Nozler @ 2500 fps.
If I were in your shoes (you already have the dies) and wanted to play a bit with a round then I'd do the AI version. It'll give you plenty of smoke and will be very easy to form cases with.
Best of luck to ya!
Dober