You will get many, many varying opinions on this topic, so consider mine just one among several. I've made roughly 15 trips to Africa, however, so I have a bit of experience. On my most recent trip (2022), also spending some time on a game reserve, I took only my 24-120 and my 200-500. Were there times I wanted more reach? A few. Were there times in late evening when I wished I had a faster lens? A few. But, I am not afraid of a little noise in order to get a shot, so I simply increase the ISO. Plenty of ways to fix that in post.
In my opinion, you will get the most benefit out of a second body, having each body with a lens mounted and ready to go. Things happen quickly on a game drive. You might spot a rhino 200 yards out and want the reach of your 100-400, then right around the corner encounter a group of lions 30 feet from your vehicle. I had these exact experiences during my last trip. The animals will not wait around while you change lenses. And, dust is everywhere in most of southern Africa.
Plus, when people think "Africa" they immediately think "big telephoto for great distances." Africa is much more than that. The 24-120 will let you capture the glorious African sunsets, vast landscapes and interesting closeups. It is also ideal for portraits of the beautiful, highly interesting native peoples you will encounter (always ask if it is OK to take their picture). I personally would not consider a fast prime to be a better option; what focal length to choose? I would not trade flexibility for another f-stop.
I used to take several lenses on such trips that I might "need," only to find out that I rarely did. Every extra pound matters when you're traveling long distances. IMHO, a second body is more critical than multiple lenses, both for the reasons mentioned above, and in case of mechanical failure. You might also consider a good monopod and gimbal setup, which is easy to use in a vehicle or on foot, and helps compensate a bit for "slow" lenses. With the two lenses you have, plus TC, plus the option for DX, unless you are strictly interested in birds, you will have most everything you might encounter covered pretty well.
Enjoy your trip. Africa is wonderful on many different levels.