It seems nonsensical that BA don't oper F down to GIG.

Many years ago I did an open jaw to GIG via GRU (admitedly in CW because they said F was 'closed'), flew AA up to JFK via MIA (Y op-up'ed at gate to J for GIG -MIA) and then the old R from JFK to LHR (only BA flight between the two that day due to snow in London).

If they used to fly to GRU and then shuttle onto GIG, couldn't they do 4 class to GIG and shuttle onto GRU some days, and 4 class down to GRU and shuttle onto GIG other days?

I'd recommend flying the GIG-GRU leg, and also had a good experience with TAM at the time flying to IGU and back in same day.