From an Alaskan's point of view I'd say he's well equipped as is except that the stainless 45-70 would be a good addition to his existing battery.
A 308 with a good bullet, 165-180 grain Partition or Barnes TSX of the same weights will do a lot of good work up here. As was mentioned I'd rather see him with a rifle he shoots well than one he is uncomfortable with due to recoil or unfamiliarity.
My wife has killed moose with a 308 loaded with 150 grain Barnes TSX bullets. They worked fine and are what my wife's 308 likes. A close friend and his wife keep the freezer well stocked with moose, black bear and caribou. He shoots a 7mm08 and she a 300 Savage. Neither seem concerned about being under gunned for any species of bear. An acquaintance's wife has killed several moose and brown bear with a 300 Savage.
Lots of lifetime Alaskan hunters go afield, taking moose and brown bear with their 270s and 30-06s and do fine. A good bullet, well delivered, from any of the above listed rounds will kill anything in Alaska.
If the move to Alaska is an excuse to buy a more powerful rifle, and it's something he can shoot well, then the 300 or 338 are good choices. They are not, however, necessary with the guns he already owns.
I would be very inclined to encourage him to have a handgun. It would be hard to beat a four inch S&W 629/Mountain Gun or a Ruger Redhawk four inch in 44 magnum loaded with a cast bullet 250-300 grains. There will be times, fishing, in the tent at night or dressing and packing meat when a revolver close at hand is a comfort. It's not always convenient to have a rifle or shotgun in hand all the time.
These are just the opinions of someone who has lived and hunted in Alaska for almost two decades.