This posted while I was answering. I've been researching that issue because I'm aiming to go to Seychelle or Mauritius to go fly fishing within the next couple of years. You can fly SA Air from Atlanta but if you're a real Alaskan, you have a ton of Alaska Air miles. Qatar Airways is a mileage partner with Alaska and there's a non-stop from Seattle to Doha and a non-stop from Doha to J-Berg. One nice thing about Qatar Air is that you can extend your stopover for a day or two. I did this with Emirates a few years ago on a trip to Europe. It really helped with jet lag.
I don't now what would happen with rifles on a 24 hour lay-over but if Qatar Air is pushing that layover, I would imagine that they could handle that situation. And that one day means you don't immediately start hunting after a 15 hour flight and you don't know what's up. There was a recent podcast on Big Game Hunting about newbies to Africa. The guy who was interviewed described landing at 4 in the morning after 30 hours of flight. He drove 5 hours to camp, sighted in his rifle and went hunting. Lo and behold, he gets a great kudu that night but he says he was half out of it from jet-lag and exhaustion. Seems to me that getting a layover in Doha and spending a night in a hotel to get a good night's sleep and adjust to the time zone is a small price to pay to be able to fully appreciate the experience.