it seems like it's not how the contract was written

Also, on previous commercial disputes, these french studios and media outlets held an all or nothing stance, often asking to be completely removed from the offering of ISPs if they can't get the money they want.

That's their way of getting what they want ; and as the ones who support this kind of move put it (so pretty much THEM) : you have the right to contract and do what you want with what you own

It's embarrassing enough for Sony or ISPs, it's highly visible to consumers, so they will not accept a middle point like you describe.