Good luck, I would probably go with the .223.
You say you don't want an exit wound. I guess you plan to keep the hide or something like that? Consider your choice of bullet and how it expands in the event you do have an exit wound.
Also don't know if you have thought about it, but from what I hear about it... Mountain lion is supposed to be pretty tasty, white meat that is sort of similar to pork. Mountain lion, the other other white meat.