I don't think it is as bad as you suggest, as through flights to EZE are only on Mon/Wed/Thu/Sat . The other days go to Rio instead.
Have you rung up and asked for the flight bookings to be joined together manually, and still been asked for the extra miles ?.
I have done a LHR -GRU and GIG to LHR open jaw in First using miles in the past and it only cost me 210,000 miles with the AMEX voucher.