I'm not any pro when it comes to mountain lions but having a background in hounds I've had the opportunity to sit in on many conversations about them. Sometimes for hours. Many hound hunters use .22M for them. Why? They don't really want to knock em outta the tree with a bunch of life left in em so they chest shoot em with the little rimfire and they bleed out in the tree. Takes a minute or two before they drop but they are dead when they hit the ground and it saves wear and tear on the dogs.
Now if I was hunting them without dogs I'd obviously want something bigger but like the OP I'd want to save the hide given the chance. My vote goes for cast lead in a .357, 44, .45LC or even a .30-30. No HP's but something with a good shoulder and a big ol' flat nose on it. Of the guns the OP mentioned I'd go with the 7mm or .308 and work up a good FP cast load of moderate velocity for em.