Actually, the .338 is all that... the 210 gr. nosler partition can be loaded to 2900 fps plus, which shoots every bit as flat as your .270, and is nipping on the heels of the 7mm Mag. The 250 gr. matches the trajectory of the 180/30-06. The 225 gr. might be a good compromise.
Yes, it's a moose, 1200+ lbs of lean muscle. Lot's of threads on here about how moose can been easily killed with a .30-30, .25/06, .243, .22mag, etc. Well, so can an elephant, but that doesn't make it an appropriate choice.
The .338 isn't a cannon... it's a medium bore, designed to take medium sized game. It's also the most versatile calibre if you truly have the opportunity to hunt all the game of North America. It's an ideal choice for anything from elk on up.
And right at this moment, I don't happen to own a .338, I have a 9.3x62. I am looking though, just not finding what I want.