Yes to the first part. Put your site behind a login wall that requires users to sign a contract to that effect before serving them the content... get a lawyer to write that contract. Don't rely on copyright.
I'm not sure to what extent you can specify damages like these in a contract, ask the lawyer who is writing it.
Contracts generally require an exchange of consideration (something of value, like money).
If you put a “contract” on your website that users click through without paying you or exchanging value with you and then you try to collect damages from them according to your contract, it’s not going to get you anywhere.
The consideration the viewer received was access to your private documents.
The consideration you received was a promise to refrain from using those documents to train AI.
I'm not a lawyer, but by my understanding of contract law consideration is trivially fulfilled here.