Sadly you're between a rock and a hard place ;-{
Reflowing or reballing doesn't fix things. Replacing the GPU is your only option in saving your logic board. Not may people have the right gear to do this.
Replacing the logic board I think is a good option but it will also limit you! You see this series had two versions of the board one with AMD Radeon HD 6630M and the other with Intel HD Graphics 3000. I would recommend you stick with the Intel HD Graphics 3000 versions of the board. The graphics are not as good as the AMD Radeon HD 6630M but you'll less likely have a GPU issue.