When the .243 first came out, it was touted as being the only rifle anyone would ever need, for any sized game. A lot of guides read the reviews and bought them, using them to back up clients who weren't, perhaps, the best shots. The .243 worked admirably on the ungulates, but it was a different story on the carnivores. A wounded bear, full of adrenalin and a bad attitude, was a distinct overmatch for the little cartridge, and the .243 experiment was shortlived. There is no substitute for a large frontal area on your bullet, when you are trying to put down something with evil intent in its heart towards you.
Having said that, I know of one elderly gentleman who swears by the .243. He has probably taken more game with it than I will ever take with all my other calibers combined. It probably helps that he is a wildlife biologist who knows animal physiology backwards, and is also a great shot from any position.
If you shoot a minimum of 2000 rounds practice at the range, using heavy-for-calibre bullets, from all positions, including some awkward, off angle, uncomfortable ones, and you can put all your shots into 4-6" at any range up to about 150 yards, and know how to hit the vitals of the game you are hunting from any angle, then by all means use your .243 for moose or any other ungulate. If you can't do all of the above, get and use a .30 caliber or bigger, and practce with it until you can.