In reply to PaulC:
I've just got back from an independent trip via the "alternative" south route (leaving directly from Terskol and avoiding the cablecar system).
We used Elbrus Tours to organise transport to/from the airport, OVIR registration and a night of hotel accommodation in Terskol at the start and end of the trip.
I would advise using a company which can provide the service of an English speaking representative (even if this is just via phone) as communication can be a problem.
The rest of the trip we organised ourselves and either camped or used huts for accommodation.
Message me if you need any more information.