If it's already dead, then why not go the cheap route and clean it with distilled water (the white crystals are acidic, so may be conducting due to air moisture), forcing the water underneath the BGA with a lightly pressurized water jet from syringe or similar.
Then let it dry and if residue persists, clean with some isopropyl alcohol (which is useful to have around in case you don't yet have it).
If it works, it was a cheap repair. If it doesn't, you can be confident you tried what was within your capabilities and didn't waste a lot of money.