Whether a .223 is a good choice for cougars depends more on how you plan to hunt than on that caliber or the bullet in it.
.223 will kill cougars well, no doubt, especially with frangible bullets if the cat offers a shot into its chest cavity. But it is not only how easy lions are to kill but how hard they are to find after the killing shot if they run.
Unless there are dogs along to follow up the shot, wounded cougars can be extremely difficult to trail or blood trail unless you have snow. For bare ground in brush etc. my own choice is to make sure I have an exit wound and hope it bleeds enough to follow. On moss or duff, a soft footed animal like a cougar leaves nothing to trail unless there is blood.
My son shot one with a 180 grain Core-lokt in 30-06 and we had a hard time finding the dead cat on moss in timber. The lion was hit mid ribs, angling back to exit behind the last rib. Intestine plugged the exit hole so no blood came out. After the spot of the hit, we only found a few drops in his first jumps, none after that. We found the cougar dead within 100 yards as I recall, by circling the direction it ran. In thick brush we might not have found it.
A list of factors that drive my choice of lion rifle start with tack driving accuracy to thread a shot through a small hole in brush at a cougar sneaking in on the call. Such cats often give little choice in selecton of bullet placement, so a robust bullet up to less than ideal shot angles is next. An exit hole to leak more blood is important, but not a big hole to damage the hide too much. Those last two preferences are in conflict.
For now, my compromise on all points is a 95 grain Nosler Partition in a .243, and I keep planning to try some solid copper bullets. Fairly tough bullets in moderate velocity .24, .25, 6.5's and 7mm's seem to fit our style of lion hunting. They are cheerfully more than needed, and with the right bullets give consistent exits (I.e. probable blood trail) without massive damage to the hide.
This post dittos Bobcat's comment, only he said with fewer words!
Good luck with whatever you use.
