Hang on... if the ticket was in fully flexible F, shouldn't it have been changeable for fare difference only? (That said, perhaps not, as F-out, A-back ex-DXB is pricing up at $12k at the moment)...
J all the way is only about $1,500 cheaper though, so it looks as though the OP would do better claiming EU261 downgrade compensation.
(Also, while EU261 liability sits with the operating carrier, and it definitely applies, I assume that it is definitely BA that is the operating carrier as it's not actually operating the onward flight - although it was supposed to...)