Haha...lot's of options. Last time I did Rio a couple of years ago from London, was an interesting route for sure. It was LHR-AMS (Economy -BA)-MAD- GIG (Biz w/ Iberia). And it was the same routing back.

Saved me a ton of money and gave me lots of room on the long haul from MAD to GIG, even if Iberia is not the best in class. Still much better than flying in econ for that long .....and the price of the whole trip was just a $100 or 200 more than that of the direct one from BA in econ. And lounge in MAD is quite nice!

Guess, if you have time to spare and are in no rush, you can play around a bit with the schedules to find the best fit.