Kaapland or Kaap Republiek is probably the best bet (though Zuid-Afrika is also a very strong possibility). But I also see it as incredibly unlikely that the British would hand back the Cape Colony - they need it to keep the routes to India clear.
That doesn't sound grammatical.
De Kaaprepubliek (with an article) sounds slightly less off, but still weird for an independent or semi-independent country. For some reason.
Republiek van de Kaap maybe. Also: if it remains a dominion, would it even be called a republic?
My bet would be just "De Kaap". Just as the various autonomous lands of the kingdom in OTL are just called Nederland, Aruba, Curaçao and Sint-Maarten. And Suriname was just Suriname as well.
I guess it might drop the article if it ever became independent, according to the weird logic that that made
De Oekraïne lose its (on the insistance of the Ukrainian government). See also:
de Soedan, d
e Libanon. (And:
de Kongo, but I'm not sure if that was ever standard usage.) But
Kaap sounds weirder - perhaps because it's still a recognisable Dutch word.