Moose are not that hard to put down, not like an elk in my experience. However, you put a non-vital hit on one and you're very unlikely to ever see it again. Any legal combination of bullet weight/diameter/velocity will put one down as long as you don't attempt a shot that's beyond it's capabilities. If you're using a .243 with a target bullet, you better not try a quartering-to shot at 300 yards. Stick that same pill through the ribs at a good angle and it's lights out every time. That's why I use heavier tough bullets moving at higher velocities - gives me the option of taking that off-angle shot if it's the only opportunity given. Otherwise a broadside lung shot with a 300 magnum is not going to impress the moose any more than a 7mm-08 would.
Totally OT but since OP found his rifle I'll veer this one off a bit.
Having never hunted elk, but seen this statement posted many different places has me curious.
What is it about elk that makes them tougher to kill than a moose/whitetail?
Just a tougher animal in general? Thicker skin? Longer range shots?
Thanks for indulging!