OP, I can't speak to modern muzzleloaders, in-lines, synthetic powders, etc, but I can speak with a lot of experience with traditional muzzleloaders using genuine black powder. Literally thousands of rounds of patched round-ball, covering over a decade when I was younger. This means the patch has a wet patch lube on it, too.
As long as you aren't getting rain directly into the nipple/flash hole, you can leave a percussion muzzleloader loaded for years and expect it to fire. I've personally seen a friend of the family fire all six rounds out of a Remington cap-and-ball that had been left loaded for something like 3 years. The chambers were sealed with lube at the front and with the tension-fit of the caps on the nipples, and it worked perfectly.
Granted, this was in Nevada, and may not work the same in the humidity of Georgia, but you did ask about a Western big game hunt. In camp or on a back-country hunt like you describe, the only thing I'd do would be to pull the cap off, and even then only as a safety measure. But once I loaded the charge and bullet, it stays loaded until I shoot it. If you're worried about an uncapped/unprimed rifle having the nipple or flash-hole exposed, you can rest the hammer down on a piece of leather to keep rain out of the nipple. Tape would probably work well too. Leaving a piece of leather over the capped nipple is also a safety measure for those in the field with a capped gun - the hammer is just resting on it on half-cock, and you just remove the leather when you cock the hammer to fire. The leather is enough of a cushion to keep a hammer-strike from igniting the cap.
Speaking of rain - if I were in a rainy environment, I'd be even more likely to leave the gun loaded, rather than try to load it in the rain. The powder is completely out of the weather inside that barrel, but pouring it in the rain? Priming or capping it in the rain? I'd load it at the tent or under a tree, and keep it loaded until I fired it at the animal, even days later. I've shot in pouring rain in a couple of competitions, and didn't have a single problem with misfires. The only thing I can remember doing differently is firing a couple of caps on an empty gun before the first shot to make sure the bore was dry of any moisture, which is exactly 1 cap more than I would normally shoot to make sure there was no oil in the nipple. Flintlocks can be a little more challenging to keep reliable in rain, but you can mitigate this if you know what you're doing.
Keep in mind with all of this, that the pioneers, mountain men, and natives who used these guns didn't walk around waiting to load their guns - they kept them primed in all conditions. And they'd leave them loaded for weeks or even months at a time - powder and shot were precious. When it got rainy, they'd simply cover the lock with a piece of leather or their clothing, and keep hunting or fighting. They were plenty reliable.
I grew up shooting these guns, and something I've noticed is that the hunting community who try their hand new with muzzleloaders tends to see them as extremely unreliable, finicky, etc. This is generally just a lack of experience, and hearing some horror stories, likely from similarly inexperienced shooters. They won't be any more finicky in the field than they are at the gun range during practice. They won't have ignition problems in the field that they don't have in practice. Fire a hundred rounds in practice, and you can use that data to figure out what that gun will do in the field.
With all this said, I do also have to suspect that all of what I've experienced with traditional muzzleloaders and their reliability is simply even better with modern guns, their actions, and their ignition systems. In terms of reliability, you'll be absolutely fine leaving the gun loaded on a multi-day hunt. Just take a couple of fouling shots on an empty gun with your primers/caps to make sure there's no oil down there, and then load it up for the duration of the hunt.
This!!!!!
