From my experience, a problem caused by the BGA-Package that can be solved by reflowing the chip manifests itself rather differently: Artifacts once the card gets warm, sudden crashes with otherwise normal operation, etc.
The problem you are describing sounds more like the symptom of a defective GPU-Chip.
This could be caused by a rather nasty effect called electromigration: Atoms in the coducting elements are pushed out of position by the electrons moving through the conductors and pile up in other locations where they cause a short-circuit. Normally, Chips are designed in a way that this effect doesn't destroy them too quickly, but when operated at high temperatures, or applying excessive voltage to reach the desired clock-rates (which can happen even without the user playing with overclocking just because the manufacturer decided to increase the voltage until the chip works instead of tossing a bad chip), then it is entirely possible to see a GPU die within one or two years.
The Pentium 4 Northwood-Cores were particularly vulnerable to this effect, leading to the "Sudden Northwood Death Syndrome".
Unless there are some other hidden problems in your card, which I doubt (basically everything except for the RAM and the Power-Supply is integrated into the GPU today), there is no way to repair it.