Its not worth the risk.
Can it be done., yup, it can, but should it? probably not.
I saw a smaller bull elk in idaho take 2 243 rounds to the ribs and run a long way. When we got a second chance, the father of the boy said if he does not go down, I will shoot with my rifle. (a .280 with 165 or so partitions) The boy shoots again, the bull is obviously hit, but does not go down, the dad shoots and with in a few seconds its obvious its all over but the dying.... I left them and went back to get the others to help pack out the meat, and when I got back, the boy was disconsolate, he was shooting 100 grain federals, and the first bullet hit leg bone and ribs, and did not enter the chest, the second shot broke a rib, but poked only minor holes into the chest. the third shot, almost dead on from the front, had hit the shoulder bones and broken up. the dads shot was a near perfect heart lung shot.
The ammo was 100 grain fed premium boat tail ammo, which was probably the wrong ammo choice, but it was a good ammo with a good game bullet. The problem when the shot thru nerves, bad luck or what ever hit something more solid than muscle, they did not have the mass to continue on thru the animal.
On that same trip, I saw two .257 roberts shooting 120 grain partitions and a 7.mm-08 with 140 partitions harvest elk with no problems, and in all likelihood, no more recoil. I used a .348 Winchester, because I wanted to use the gun it came in....AND I limited my shot to about 120 yards, AND I was culling. On other trips I have used a .280 and a 7x57 handload.
If your friend is recoil shy, buy some or load some reduced recoil loads, and let him do all his practice with that. then take his rifle and resight it in with full power loads, he will NEVER feel the recoil when he has an animal in his sights.
While you asked for first hand information, I was not the one pulling the trigger on a .243 on Elk, but rather a by stander. I would not use a .243 on elk unless I had NO other option.