Hi Lindsay,
I have just returned from a one month trip to South Africa, Tanzania and Kenya. This was my fourth birding trip to Africa in recent years. I travelled with my 600 mm f/4 II and 5DIII (with battery grip) fully assembled in a Kinesis lens case. The lens case I use is actually the model recommended for the 500 mm f/4 but it is a perfect for the 600 mm f/4 II plus camera body. There is absolutely no problem with dimensions; it is smaller than lots of other carry-on luggage and the total weight was just on 7 kg, the maximum permitted. You are also permitted to carry a laptop. I had no problems with Qantas, SAA, Kenya Airways and Precision Air and my carry-on gear was only weighed once. My other photography gear was shipped in checked luggage; 600EX-RT flash with Better Beamer, 1.4x III Extender, battery chargers, photography vest with small selection of Allen keys and adjusting tools, spare batteries, memory cards etc.
Jill travelled with her 300 mm f/2.8 II + 1.4x III extender and 7DII assembled in a carry-on backpack. In checked luggage she took a Gitzo 5xxx series monopod, 430EX II flash with Better Beamer, battery chargers, laptop power supply, power plug adapters and other smalls. Absolutely no issues with her kit.
I strongly recommend you take your 500 mm f/4 + 1.4x extender. In parks were big animals are present you will be restricted to shooting from your vehicle and not permitted to drive off-road; this limits the choice of shooting angle and range. Even with my 600 mm + 1.4x III I was often frustrated by not being able to get close enough to small birds.
I hope this advice helps, best wishes,
Ian Wilson