There is no cheat code here.

No one said there was one. This isn't about looking for way to break the law and get away with it, this is about the people who want the law to work a particular way not understanding that it doesn't actually work that way.

The output of LLMs is illegal.

No its not. There is no way in which the output of an AI can be illegal. All can be proven is that the various providers did not pay for the various licences but that's not the same as saying the output is automatically a crime, if it was then we'd not even be needing the case. The law is incredibly vague in this area.

Sam Altman's goal in creeping around Washington is to try to get laws changed to carve out exceptions for exactly the types of stuff he is already doing.

Yes and that's a good thing. Think about it for 15 seconds. If it weren't for people like him AI would be limited to the mega corporations who can afford the licensees, we don't want that, we want a AI technology to be available to anyone, we want AI technology to be open source. None of that can happen if the law does not change.

You seem to be under the impression there is some evil sadistic overlord here trying to force artificial intelligence on the world when it does not wanted, but nothing could be further from the truth, if anything artificial intelligence is being developed in a way that is surprisingly egalitarian considering the corporations that are investing in it, and vague unclear unhelpful broken copyright law is getting in the way of that.