"...hoping I could get away without spending 80 dollars a box on ammo..." You can. Ammo that uses premium bullets runs that kind of money. You don't need premium bullets for Bullwinkle.
Shooter's Choice wants $28.95 per 20 for Remington 180 grain .300 Mag. Bullwinkle will be just as dead.
You will have to try a box to see if your rifle shoots it well though. A 2 or 3 inch group at 100 is fine for moose. Sight in on a bench then practice shooting, offhand, at a 9" pie plate, at 100 yards until you can hit it every time.