You have to consider where you are hunting and when. If you are hunting in the Yukon, Northern B.C. etc, during the rut, then you could be calling in some very large moose. For that you would want a minimum 180gr bullet. If you handload then you could also go to a 200gr or 220gr, particularly if it is a hunt during the rut; the shot distances should be 75 yards or less (when calling in bulls).
If you get a shot at a big bull make sure of the shot angle. It's real easy to hit heavy bone and fail to reach the vitals, when using a lightweight bullet, and personally, I consider the .30 cal. 180gr light for moose. If you're going to use a 180gr then handload a PREMIUM bullet. Start with the Nosler Partition; other options would be Speer Grand Slam, Nosler Accubond or Barnes TSX.
I quit hunting moose with my .30-06 back in '88, when I hit a large bull with admittedly a marginal shoot angle. I was fortunate to kill the bull the next day , and then I made sure of my shot. After the post mortem examination I realized the .30-06 is marginal against large bulls, unless the shot is placed just right and I bought a .338WM and I have hunted with it since*. In my opinion the .338WM is the ideal moose caliber (with 250gr bullets).
On the other hand lots of guys get moose draws in the farming areas of central Alberta, and the majority of the bulls these guys are hunting are "bullwinkles", only 2-3 years old. I don't think there would be much of an issue shooting the 150gr or 165gr bullet in that situation. Just hit 'em with a broadside, double lung shot.
Edit:
I also meant to say that if you're hunting late season moose then you're likely to be hunting cutblocks and then shot distances could be longer. You would also be much more likely to be taking 200 yard shots. I think the 165gr or 180gr bullets would work then.
Also it is a myth that the 150gr bullet has an edge in long range trajectory. Ballistically the .30 cal 165gr (boattail) is the optimum wieght for long range shooting, but generally, moose hunting isn't about long range shooting anyway.
*In 2006 I used a .416RM