Most people will use Cash plus Miles and book Business class. The cost each way to Brazil in business is 18,750 miles plus £128 so if you have 37,500 miles you can do this.

If you have trouble finding business availability on the normal GRU routes have a close look at the TP flights LIS-SSA and other points north - take a delightful stop in somewhere like Fortaleza or Salvador de Bahia then go to Brasilia from there.

As for dodging the APD, you could consider Easijet from LGW or even Ryanair (ugh) from STN to Oporto and an award on the direct TP service OPO-GRU. There's no APD coming back so if you book two singles you can get an award back to LON via whatever routing.