Any quality 180 gr bullet in a 303 will put a moose down with a decently placed shot.
In fact, from the end of WW2 to the 80's probably more moose were shot with milsurps, especially Enfields than any other gun. I have shot them with 8mm Mauser, 303 Enfield, 30.06, .308, and more recently a .300. The only benefit of the magnum is when we are hunting cutovers, and shots may be 2-300 yards. But my first moose with an 8mm Mauser was 340 yards and I was 15 years old - took a 2 foot holdover, but I got her.
Don't be fooled into thinking that moose need big bores or magnums to put them down. A chest shot and they die. Not always on the spot, but they do not run like deer. They will walk until they feel safe, lay down and die.
Good luck with your first hunt.