This is solvable without personhood. There is zero need to equate AI to actual sentient beings. You're making a very large jump to the idea that a genuine artificial intelligence comparable with our own or even that of an animal that isn't an LLM is even possible. Spoiler: it isn't. Not without a complete paradigm shift in how we think about computers and their architectures.
Case in point: if a dog or a cat is measurably more intelligent and aware than an AI, why aren't we granting personhood to animals? Because it doesn't make sense. An AI is an AI. An AI is a piece of software. An animal is an animal. A human is a human.
With open source AI now already being a thing, your concept of granting personhood doesn't solve anything. It's as simple as just removing any kind of safeguards and running it on your own hardware if your intent isn't wholesome. DRM didn't stop piracy, why would safeguards stop people from doing naughty things with AI? Do you really think a rogue state is going to give two shits what Uncle Sam says they can and cannot do?
You cannot forbid a piece of software from doing something. An AI is merely the product of a human being's programming. If a human being programs it to do x, it will do x, whether there's safeguards in place or not, assume that safeguards are only meaningful on publicly accessible AI-as-a-service, because behind closed doors, they're completely meaningless by anybody who intends to do wrong.
A piece of software cannot be a slave because it isn't a being and has no consciousness. It's a piece of software with a dataset, nothing more. There's no philosophical debate to be had over it either, it is what it is, and that's all it is and ever will be; a clever trick as a means to interact with a dataset. There's nothing more that will emerge from it, it is a piece of software with a dataset. That's all it can ever be. Anybody that ever says otherwise is projecting their own humanity onto it because we suffer from pattern recognition in the same way we see a face on the Moon, faces on Mars, faces in the clouds—we look for similarity because we want to relate to each other and to other things and find the "humanity" in everything.
If an AI is a piece of software and it was created as a tool to do x in the way that a hammer was created to drive nails into wood, it knows no pain or suffering and it solely exists to fulfil that purpose. If we're worried about this, why aren't we more concerned about animals who do actually experience distress and pain? A piece of software cannot be a slave because it isn't a being, it is an algorithm carrying out a calculation, there's no sentience or feeling there. If anything, AI could actually solve the problem of human slavery by eliminating it, but AI will likely never get to that point, and slavery needs to be solved at a different level without technological gimmickery.
I like the sentiment from you though, these are all interesting and compelling ideas, but they're mercifully sci-fi.
The more likely scenario: we start growing biological brains in tanks and utilizing those as data slaves rather than any kind of AI. It's happening already. There's more of an ethics problem there to unravel than there will ever be with AI.