There is no way to do so without making "South Africa" be both larger and colonised by a different country. Let's say Spain for the sake of the argument.
So Spanish South Africa develops like Latin America, with the majority of the population being mixed-race, although probably more mixed-race since there will be far more natives because of the lack of epidemics, even after the Spanish remove millions of them through enslavement or feeding them into gold/diamond mines (natives will "help", possibly through a Mfecane analogue).
This South Africa will be far bigger than modern South Africa, but let's limit it to all the lands south of the Zambezi River plus with a line drawn as the modern Angolan border. That's probably the limit a state in that area can get without regional secessionist tendencies, especially because the bigger the future country, the harder it is to get a large mixed-race Hispanic population established to the detriment of the natives. To conquer this land, something like the Brazilian bandeirantes are active--I'd say that's particularly plausible, since South Africa has lots of mineral wealth as well as slaves to capture. For the more organised states in the region (like Monomotapa), Spanish expeditions aimed at them end up successful and their people broken and incorporated into the Spanish Empire. African conquistadors, let's say, although they'd have to do better than the Portuguese did in that region.
Maybe Spain has some issues of its own, and key parts of their Latin American empire never develop. Let's say Pizarro fails to conquer the Inca. And let's also say Spain is promoting migration from more of their empire, as well as sending the usual emigrants who would have become the ancestors of many Peruvians/Bolivians.
Now, this country gains independence in the 19th century. It's early years are turbulent, but skilled dictators manage to create a sense of stability in the worst years and eventually a corrupt but functioning democracy evolves. Even though this South Africa is poor compared to our own South Africa, it still manages to attract immigrants because of few restrictions, active recruitment, and worse conditions than OTL in the New World for immigrants (more anti-immigrant legislation and just weaker countries in general). Turmoil in Europe and lack of solutions to dealing with emigration creates more prolongued than OTL migrations. Many South Africans as a result have German and Italian backgrounds, and South Africa has one of the largest Jewish communities in the world.
In the 20th century, the population grows rapidly, much as Latin America's historically did, as medicinal advances spread. Good healthcare policies and leaders hellbent on exterminating common diseases lead the region to eliminate smallpox at a rapid pace and make great strides in the elimination of many other diseases. But in the mid-20th century, a new dictatorship emerges that believes strongly in natalism (moreso than most Catholics) to ensure South Africa's population continues to rapidly expand to help South Africa be able to dominate world affairs. Although democracy returns within a few decades, this era of South Africa is marked with continued and artificially high expansion of the population. To deal with water issues and such, the government invests strongly in nuclear power to power desalination plants--even now, South Africa is known for leading advancements in desalination technology.
Flashforward to 2016, and South Africa is the most populous country in the world after China, with over 300 million people. The new governments have done a great effort to get the fertility rate down, and South Africa is now considered a model for how countries should approach limited births.
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So did I do it? I personally think this is a Spanish South Africa wank--realistically 150 million or so is the most even that expanded South Africa could probably expect.