Both your 308 Win and 280 Rem will work well on moose with "hunting" bullets of 150-180 grains and 140-160 grains respectively, placed properly in the vitals.
Use either or both, but I would recommend the one you shoot best, with the ammunition or load that the rifle prefers.
In 35 years of hunting and guiding moose, I have not had to shoot one past 220 yards. I have used the 30-30, 308, 30-06, 6.5x55, 6.5 Creedmoor, 270, 280, 7mm Rem Mag, 7mm STW, 300 WSM, 300 Win mag, 338 Win Mag, 358 Win and 375 Win, over the years, with a variety of cup and core, and monolithic bullets in handloads and factory ammunition, all with good success.
My Dad, grandfather and great grandfather have guided moose hunters (Cree guides) for a combined total of 135 years, from the Watertons to Terminus Mtn, and used the 30-30 almost exclusively for their back up and own hunting needs. My grandfather even used a 22LR when on the trapline to harvest moose; "tickled them in the ribs" (or as stated above, "let the air out"), sat down rolled a smoke and then walked over in about 15 minutes to find his moose dead after it had wandered off about 50 yards and laid down to expire. Grandma even used slugs in her single shot .410 when out camping. Point is, moose are not overly tough to kill.
Many big bull moose in the rut, when hit in the vitals, do not instantly show any reaction to the shot, or if they do, they may hunch slightly and take a few steps, stand still for a moment, then go down in some dramatic fashion. Best to be prepared for a quick follow up shot, but in many instances it isn't necessary.
My personal favourite hunting bullet is the Nosler AccuBond. It is a top notch on-game performer and has is the most consistently accurate bullet in all the rifles I have used it in to date, and also the easiest to find good loads for. I have taken antelope to bison with it, in factory ammunition and handloads, from the 270 Win up to the 376 Steyr, to date.