Travelling from UK to Russia

I intend to use my South African passport to exit UK because South African passport holders are entitled to enter Russia without a visa

What passport you'll use for entry to the destination does not affect what passport to use for exiting the departure country.

Will Border Control at UK port of exit permit me to leave UK for Russia using my South African passport?

If the UK did border control on exit at airports, no, you'd have to use your British passport, as with the South African one they'd look for an entry stamp and visa or BRP, and suspect you of being an illegal immigrant if not shown your British passport.

However, since the UK doesn't do border control on exit at airports, the issue is moot.

At bag drop, though, you should present both passports, explaining the British one is to be used for UK API purposes (advance passenger data - the staff will understand), and the SA one for the passenger data for Russia.

For this reason, I recommend you not to use self-service bag drop (if available) even if checking in online, as API forms online normally only allow one document to be used. Since Russia also does API, your UK passport has to be used for UK API and the SA one for Russian API. As such, bag drop staff may have to correct the data.