Exporting of firearms and even gun parts out of United States is controlled by the Department of Defense, not the Department of State. There is a special permitting process that must be followed. I know this because I've been considering getting one of these permits for my own business, I have spoken to several lawyers over the years about the process. I have also seen what happens when this process isn't followed, at a minimum the loss of the merchandise, and more often than not a criminal prosecution on the federal level.
This is best handled by an exporter, a company or individual that already has a DOD permit and can apply for a dealer in South Africa to take possession. Some companies have affiliates in both places so they are the best ones to streamline this process. I know there are sporting goods stores in Johannesburg /Pretoria that handle this type of thing, I'd suggest simply to check with them to see if they can help. Your professional hunter should know who to call or certainly will know someone that does.
This S.A. dealer must be willing to do the following...
Once in S.A., the firearm will then be subject to permitting to the new owner by the South African government, and the firearm must remain in the posession pf the dealer until the paperwork clears, a process that can take years. I think My PH told me it took him 3 years to get his last rifle.
These forms can be found here on the Professional Hunters Association of South Africa's website...
Sect 16A PH Status
What surprised me about this whole process was the "motivation" that the pH must write, a document describing why this firearm is absolutely necessary to his collection and why other firearms in his collection would not serve the purpose of this new one. Thank God for the Second Amendment.
Hope this helps...