Depends upon the bullet construction, bullet velocity and where you hit the cat. Some guys might have experience with the real light high velocity .243 bullets, I don't so can't offer much advice about them. Hit on the shoulder I would be worried about a fast stepping lightly constructed bullet doing a lot of damage to a bobcat. I say that because of some close range .22-250 experiences. With the .243 I've shot several cats with the older 87 gr. Hornady BTHP, quite a few with the 85 gr. Sierra HPBT GK, one with the 75 gr. V-Max and a few others with both the 100 gr. Hornady Interlock soft point and Speer 105 gr. spitzer soft point. All in hand loads running from 3,200 fps to about 2,900 fps. I've seen everything from silver quarter size exits to softball size exits. I have a cat on the wall shot with the 85 gr. Sierra. That cat was shot quartering on right on the point of the shoulder, the entrance hole was .243" but under the hide the shoulder was pure jelly. The bullet exited just in front of the opposite hip with a clean quarter size hole. The mount has that back leg pulled up as the cat is climbing up a rock so the hole is covered and was never a problem for the taxidermist. With luck and skill a good taxidermist can plan the mount around a hole in the right spot. I'd use a strongly constructed deer class bullet, maybe lower the velocity. In factory form one of the managed/low recoil deer loads might work pretty well. Believe it or not a shotgun with appropriate shot used at a decent range is excellent for the taxidermist. Another option for consideration. I've shot quite a few bobcats with the shotgun and sold every hide without a problem.