Dr. J, There's a thread started on Aug. 13, 05 titled Caliber for Calling Cougars that addresses this topic. Most of us leaned toward accuracy as most important, with wallop secondary. The reasoning is that since cougars are usually so sneaky on their approach, you may have to shoot at a small piece of cat through a hole in the brush, so need the accuracy. But once hit, cougars are generally easy to put down with most any centerfire, so it doesn't take a boomer. I favored the 6mm/243 and some like the fast 22's like the 22-250. My son shot a called one with an 06 using a 180 grain Corelokt because that's what it was sighted for at the time, right after elk season. The heavy bullet didn't make a very large exit hole, which was nice.
As to bullet, again, all but the lightest varmint bullets would do. One way to look at it is that quick fragmentation of a light frangible bullet inside the chest cavity should drop a cougar on the spot with no exit hole in the hide. I prefer an exit wound because where we hunt cougars, it is virtually impossible to track one unless we have snow or a blood trail. If I hit one, I want to find it on an endless carpet of moss.
For that reason I have carried 95 grain Nosler Partitions for cougars in my heavy 6mm, but haven't shot one with them. It is super accurate, and should shoot through yet make a fairly small exit hole in the pelt. We've shot several deer, bears, an antelope and a bull elk with the 95 grain Partition, though I don't recommend it for anything bigger than small deer or antelope. I used 75 grain Hornady hollow points for coyotes, and I'm sure they would do for cougar though I want a little stouter bullet to insure an exit hole. I recently picked up a .243 light carbine to use on cougar, and am trying to make a bullet choice myself. I'm thinking of experimenting with something other than the 95 grain Partition, likely an 85 or 90 grain bullet. I'm looking at the 90 grain Sirroco bonded but need to find out how big a hole it opens.
In any caliber larger than .24 I'd simply go with the most accurate bullet I could try that would not blow a huge hole in the critter on exit. Hornady Interbond should be good if it doesn't open too much. Any partition should do. Any of the standard bullets will kill the cat, but many will expand so much they damage the pelt badly. The Sierra Game King is often very accurate but sometimes open pretty big holes. Accuracy, accuracy, accuracy.