Well, the simplest test is to heat up the gpu to somewhere around 150c (not enough to melt solder) and see if it works afterwards. If it does the gpu is done and needs replacing, if not there may be hope.

Of course you can try testing the ram and do other troubleshooting first to avoid doing potentially unnecessary heat cycle on the gpu, but honestly imo the probability of dead gpu is very high.

The board had a slight curve to it that was caused by the cooler.

I decided to try a reflow of the core and memory, so I mounted the card onto my preheater and used weights on each end of the card to flatten the board.

With some flux applied around the core and memory, I reflowed the components.

After allowing the card to cool down and removing the weight, the curve in the board is almost gone.

I tested the card and it's working.

I have been able to install the driver in Windows XP and ran 3DMark2001 SE with a score somewhere around 18500.

The only remaining issue I have is that the cooler fan is broken and I'm not sure what to use as a replacement.

This is what the cooler should look like but on my card the plastic fan is completely broken and unusable.
I also had to remove the copper plate on the memory to do the reflow and I don't have replacement double-sided cooling tape to re-apply the plate on the memory.

Well, by actually melting the solder you lost the ability to diagnose if it is BGA issue or not. Heat will almost always make it work, the only question is if the issue was with BGA and is now fixed, or if it is between die and substrate and then it is not fixed and will fail shortly.