By law strickly speaking you can do anything the owner of the land will get a permit for. The land owner can usually get permits for most things he wants with very little reason much of the time. It's simply a matter of asking for it and saying something like I have had predation on my sheep or chickens. This damage permit allows the land owner or his designated person to hunt or trap the problem animals. It's actually a pretty easy thing to do.
However these kinds of permits will not allow exporting the skins. For that you will need to hire a Professional Hunter who has a log book and can do the paperwork to allow export. Without the log book filled out you will not get the skins out of the country legally, nor will you be able to import them into the USA. You may be able to have the tanning done in South Africa and then have them shipped to you or collect them on another trip the next year and take them home as a curio.
Skins of all types are for sale in most bigger curio shops there. If you bring a gun you will need the name and address of the place your staying and the name of the hunting outfitter you will be with. If you borrow a gun then it's not an issue.
You would be far better off having the "friends" you have in RSA sort this out for you. They are close to the source and can make a plan with the local people for you. I'm stricktly a PH on a regerested hunting consession. We have all the permits to hunt cats and such s it's no problem for us.
Here is a caracal cat or African Lynx I shot in July of this year. I shot it while waiting at a leopard kill which was hanging in a tree.