From what I have read moose are not hard to kill but when shot they just don't know that they are dead. A guide friend of mine had a client a couple of years ago place 3 shots into a moose at 150 yards with a 300 magnum and all the shots were right where they needed to be before the moose just fell over.
For both rifles I would stick with bullets in the 180 grain range. I don't reload for the 300 but for the 06 that means around 2700 fps. I personally like the Barnes TSX bullets and with them you could drop down to a 165 grain bullet for some extra velosity. I have shot quite a few elk and a few deer with the Barnes bullets and they performed flawless every time.
Moose hunting is generally close quarters with shots no further then 200 yds and most are much closer. That being said I would go with 180 gr premium bullets for both. I too love Barnes , but try a few different bullets to see what shoots best out of the gun. Partition or Accubond may be another choice. Either way, he will probably act like you missed him. They do not act like other animals when hit. Good luck!
Biker, on the Barnes TSX you can drop down a bullit size due to their construction and still get the same effect as a heaver bullet. So instead of the 180 TSX drop down to the Barnes 165 or even the 150 grain bullet. You will carry more speed at longer ranges and bullet performance will not be affected.
The monolithic bullet have a huge advantage, they can be a lot lighter and still weight more than a heavier bullet at impact. At even 300 yds, they will arrive faster with the increase in MV and then penetrate better because they dion shed weight. A 180 gr cup and core bullet that sheds 30%, Nosler Partition, falls off to 126 grs somewhere along the penetration. The 150 gr monolithic, keep's it total weight through. The monolithic will also destroy less meat. I guessing that it's because the monolithic does't mushroom and lose weight, goes through like an improved solid. I you want to spend the monney on premium bullet's, they do work.
Elk have been killed close to roads where they could be loaded whole into the bed of a pickup. But it doesn't happen often and I'm not sure that is the experience that I long for. The type of elk hunting that I prefer requires rigorous physical exertion that can cripple someone who is not in shape. Being a college hockey player I have to be in at least fair physical condition most of the year but when one of my buddies got back from the Navy Seals he changed the way I think about physical...