I hold dual nationality - British and South African. I am due to travel to Russia flying out from Germany. I intend to enter Germany (on a visit) with my British passport and use my South African passport to exit Germany because South African passport holders are entitled to enter Russia without a visa. Will Immigration at Frankfurt airport permit me to leave Germany for Russia using my South African passport, after having come in on my British passport ?

Dont you think its best to inquire at the respective consulates?

Airline staff check to be sure you have documents necessary to enter the destination country. They are not doing passport control and may not even know which one you used to enter the country.
What I would do:
Use my SA ppt when checking in for the flight to Russia. If you must present a passport when leaving FRA (as you likely will) just present your British passport that you used for entry.
You are not doing anything nefarious; my SO who holds dual citizenship dies something similar all the time when we visit his native country. Just have both available for inspection if required in Germany.