Can I apply for a visa for Georgia while in the UK on a visitor visa for 5 months on a South African passport?
No, you cannot. South Africans DO NOT NEED visa for visiting Georgia for up to one year.
Link - Ministry of Foreign Affairs of Georgia
Or maybe you can, but you don't need to.
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@Dee Correct, Your passport should be valid for the proposed duration of your stay. No additional period of validity beyond this is required. – BritishSam Feb 21 '18 at 14:36