Black screen is almost always trace damage from a leaky battery. 3000's caps never leak and you won't be able to clean your way back to good traces.
I have seen, far to many to count, 3000's die from battery damage. Never use barrel batteries, even if they are the green NiCad ones. Only use the coin types that don't leak, unless you don't install the diode and charge a nonchargeable battery.
You will need testing equipment and the schematics. It will be a long and detailed repair for the unskilled.
Your best bet is to find someone close to you that has experience fixing amigas and take it to them. Shipping will be an absolute killer since you own the heaviest Amiga Commodore every made, drive it to someone fairly locale and let them fix it. You may even want to try a local board repair shop that can xray the board and can fix the traces.
Sorry to see that happen to you. Black screens are the worst screen, all other colors are easier to repair.
Again Sorry.
Take it from someone who has over 20 years into 3000's, had 7 3000D's even owned a 3000T back in the day. Find someone who knows how to fix trace damage and get it fixed, and never use barrel batteries again...