My idea was more about how to improve avalanche so this wouldn’t be possible. If that would be done, it has to be backwards compatible, so it wouldn’t affect your situation. Old addresses should also work. Your avax adress would then looke like C-avax1jsfdsjkfdjk… If you would copy paste that to binance, binance would say it is not a valid address (for ethereum, or BSC, or whatever chain) and the problem is prevented.
I see I didn’t really help you with that comment. I am sorry. Let’s try to do it better.
Your problem can only be solved by the party owning the private key. This is binance as you sent it to them.
They currently only support transaction to the X-Chain. If they would also support the C-Chain and do that retroactive for past transactions your funds would show up. (Regardless of any address format changes)
If I were you I would not just wait for them to support the C-Chain but open a support ticket to retrieve your funds. Maybe they will. It might also bump up the priority of supporting C-Chain if people who experience this issue report it at binance. If you look around in the forum there are many similar cases.