Lots of guys have had good luck with Delta and UA, but they are both rather notorious for very late departures and outright cancellations.
I flew Qatar in 2024, will either fly them or Emirates when I go back next year. The one bad thing I'll say about Qatar is there's usually a really long layover in Doha. But the flight attendants, while not Emirates level, were good, and I was notified at all departure points (IAH and DOH going, JNB and DOH coming back) that my rifle case was safely on-board. My best friend flew UA for the same hunt. Woopsie. They forgot his rifle in EWR, along with the rifles of all the other hunters on that flight, about 8 or 10 of them.
I wouldn't fly a European carrier from the US to Africa on a bet, at least not if I were bringing my own guns. Since you aren't, I like KLM the best of that lot. YMMV.
One suggestion I will make - overnight in Jo'burg. Africa Sky (I like that place a lot), Afton, or City (located in the airport at JNB). Probably stay at City since you've got a connecting flight. It'll give you an opportunity at a hot shower, good food, and some much needed rest before the adventure starts.
*IF* you decide to book UA or Delta, because of frequency of cancellations, I'd arrange to leave 3 days before your hunt starts. If you get cancelled, you still have wiggle room and will still get your full hunt. If it doesn't cancel, you'll have to overnight 1 night anyway before your outfitter comes to collect you. Win-win, either way.