Here is my idea: Everyone makes a private key. When they buy a song they receive the file and a digital signature by the label saying they sold it to your privatepublic key. When you are caught with a bunch of songs, you have to prove ownership using your key. Tadaa provable ownership, no blockchain, You loose the file, but still have the signature? You can download it again and all is good.
EDIT: Of course, they would sign the public key, sorry.
In this construct the key is the thing, that signifies the owner. If it's lost, there is no help. Unless the key is tied to you and an authority can vouch for you.
A digital signature from the label would be created with their private key.
What would they be signing? Your public key plus the ID of the song? They can't sign your private key, it's private.
What stops you sharing your private key and a song with a friend. Then when either of you need to provide proof, you can both show that you have the private key that matches the signed file?