I'm not really sure, or, well, at least AI in what people tend to associate with nowadays. I think a solarpunk future is a future where degrowth is a main ethos for the development of the society. AI, as it currently exists, relies on global scale exploitation of not just artists, writers, but also people in the Global South living in abject conditions and are forced to accept mentally and physically damaging jobs to help "AI" to exist in the first place. Using AI, as far as I know, also comes with huge environmental costs simply because of how energy intensive it is to run and train its huge dataset. So the existence of stuffs like ChatGPT or Midjourney is a no go, unless "human ingenuity" could do something like being so cost effective energy wise that technology like that would require way less energy to run, or somehow the colonization of the universe would happen, but for the latter to happen will neccesarily require the kind of economy that centers growth above all else like capitalism.
Imo, Solarpunk technology should be local and open source and easily understandable for communities and individual to use, the factor of being environmentally friendly is also very significant and I don't think AI infrustructure would guarantee to fullfill those requirements.
That's the reason I would never understand the obsession with nuclear power or AI techinology of some solarpunk enthusiasts. These are all infrastrcutures that requires a degree of centralization of human and resouces, which drastically constrasts what solarpunk should be about. A solarpunk future is not fully automated luxury communism, but a low tech communism that still guarantees the well being of all. Low tech does not exist simply because of accessiblity, but also the capital E Environment.
Furthermore, the basic issue of human creativity being ripped off is also a thing that should be addressed. I don't belive in intellectual property, but we shouldn't need IP to protect arists from exploitation without consents. A solarpunk world is a world that respect human autonomy, so if an artist do not consent for their works to be used in training AI, their wish will be respectfully followed through. As an artist in art communities, I don't think that many people are fond of submitting their works to an AI so that others could code a drawing, so the dataset required for AI would not be enough, thus render AI useless. That's just creatives though, maybe scientists and researchers would hold different views? If that's the case, circle back to the question of energy intensity inherent in AI technology.
That's just AI in a very specific sense though, I don't know what constitute an AI. If AIs are just machines that can beat human at chess, or to predict patterns of local weather, or regulate irrugation system for farming, or organize documents, then sure, there's no reason why it should not exist in a solarpunk future. But the existence of such machinery should be put under scrutiny. If an AI could not be operated without significant environment impact, it has to go. Impressive techonology is not just codes, it's also very real, very material hardware that possess impressive processing power, and with it, impressive amount of energy, resources, and human power.
To address your AI specificly: I think your AI is HUGE, and I'm not confident about its place in a solarpunk world. Your AI would exist in some versions of communism, but solarpunk? I don"t think so.
Eh, they don't care precisely because they believe they have the privilege to shield themselves, and their descendants from environmental catastrophe. Luxurious fallout bulkers are built right now to do just that: protect themselves and their children from iminent threats, be it mob riots or tsunami or nuclear fallout. They don't give a shit about the rest of us, and why would they? What do they have to gain from caring about us, it's just not in their interest to do that when they could afford to survive through this and we don't. They've already won the logic of the free market in a bloodthirsty, capital-centric world then surely they earned their superiority, and their place as the true, deserving heirs of the Earth reborn. Looking through their eyes, it's just natural selection.