While writing my other comment, I thought of a way that might be a little bit closer to a solution for the AI use of copyrighted material. In Germany, afaik we are allowed to make and keep a copy of copyrighted material obtained in any legal way (even borrowing, streaming, etc.) for strictly private and personal noncommercial use. As a compensation, when buying a device that can store or process data (e.g. USB-stick, phone, computer), a small part of the money goes to copyright collectives ("Verwertungsgesellschaften"). Copyright holders can be members of those and get compensation or their work via them (those collectives are also the entities that sell you a license to e.g. play music at public events).
I can imagine a similar system for AI: A part of any subscription, fee, etc. for AI could go to those copyright collectives as a compensation for the material used in training. Compared to obtaining a license once for training (like buying physical books), this has the advantage that the compensation would scale with the actual use of the AI and also the revenue of the AI companies.
I think those systems are far from perfect, but that's probably a much better approximation to fairness than what we currently have