Whichever you can hit best with. Odds are, the noise of the muzzle blast will be as effective as a hit for ending an encounter.
Anyhow, a hard cast bullet from the .357 will work. Natch, if a .357 will do, so would the .45 Colt.
If you're handloading for the .45ACP, odds are that the old standby IPSC load would work, with the 200-grain SWC--but maybe up a bit at 900 ft/sec.
It's not uncommon for a cougar to follow along and watch a hiker, but most of the time it's curiosity. As long as you're at least casually watching your six o'clock, no problem. An erect human is not shaped right to be obvious prey.
Most problems in Big Bend National Park have come about from a sudden encounter with a cougar who apparently wasn't listening for approaching humans. Mutual startlement, apparently, coupled with an absence of fear of humans.