IMO, your .308 with quality bullets will work just fine. Moose aren't real hard to kill, they'll take a bullet and just stand there giving you time to hit them again. If they do move after the shot, they just don't cover much ground, not like elk often times do.
I used my .300 WM for two AK. Yukon moose. One took 3 shots before he dropped and the other dropped before I could recover from the recoil, load, and shoot again. These moose around here are much smaller, although still have a large kill zone. Put the bullet where it belongs from your .308 and enjoy the meat.
Of course, if a man just needs an excuse for a new rifle, I'm onboard for that. Maybe a .300 WSM?