The only person I know that hunts moose with regularity (in western Canada and Montana) does so with a 7mm Rem Mag. And I don't believe he has had one get away from him in many years.
338 should be more than up to the tast but I think your shooting experience should enter into the exposure. Sometimes it is hard to judge from such a question.
How long have you been shooting?
Do you have any tendency to flinch with 30-06 or 300 Win Mag?
How closely can you keep shots at 200 yards with said calibers?
Are you the sort of person that will keep his shots to within 250 yards or so or are you looking for and ready to take a 400 yard or longer shot?
Plenty of moose has been taken with 270 Winchester. Plenty of moose has been taken with 45-70. Those cartridges are hard to recommend for someone going on a moose hunt. 300 Win Mag, 300 Ultra, 338 Win, 350 Rem Mag, 35 Whelen, 375 H&H should all be very well up to the tast and the most recommended. Provided you can take advantage of their capabilities as a shooter. As always, shot placement and bullet selection are the most important factors, assuming you're using something over a .243.
Good luck. I'd love to go on a leisurely moose hunt, perhaps on land where I knew the owner. I'm just not ready to go on a paid hunt yet.