Putting in a different Chip won't work, even if it were electrically compatible with the board, the EFI would not support it.
I've repaired one of those infamous 650M GT logic boards myself before.
Most of the time it's not the actual GPU, but a buck converter (U8900, google "macbook pro U8900" to get schematics) that's gone bad. Or, more precisely, the soldering joints that connect that chip to the board.
I reflowed just this chip and it continued working just fine. Look into that before you ruin an otherwise functioning logic board.
Replacing BGA chips with hundreds of contacts is a delicate process that requires experience and expensive equipment.
Reflowing or re-soldering the U8900 Chip requires nothing more than a cheap hot air station. I've done it and can recommend trying that.
You do need a little experience taking apart computers and likely risk damaging something if you don't have that. Other than that, it can be done, with a little patience and the right tools.