Not to discourage you from buying a new rifle but I think your .308 is perfectly adequate. The .35 Whelan would be fine too. As Whitewalrus said, a premium bullet and bullet placement are key.
ALL of my encounters with moose were and continue to be at close range, less than 100 yards. Go the the Alaskan tundra to shoot moose from long distance.
I actually scaled down when I bought my commemorative once in a lifetime 2018 Colorado bull moose rifle. Shelving my trusty Winchester 54 .30-06, I bought a Kimber Adirondack in 7mm-08. It's a necked down .308, but faster and more accurate. It sports an 18" threaded barrel and weighs just under 5 lbs. It's light, quick handling and soft shooting. Suppressed, it feels a little like shooting a .22 magnum.
Using factory Barnes VOR-TX cartridges with the TTSK 120 grain bullets, I took two shots on my bull, both of which were fatal. He took three steps and collapsed.
Fwiw, I think you should buy a new rifle if only to commemorate your once in a lifetime Colorado bull moose license opportunity. But I know you don't need some heavy beast of a rifle to reach out and touch your trophy Shiras bull.
Cheers!