It's not desert solarpunk because solarpunk implies a hopefulness for the future that does not exist in Dune.
Dune and Dune Messiah are tragic cautionary tales which warn against religion and the great man myth. It is supposed to warn the reader that religions take a life of their own and become a tiger that you can grab by the tail, but can never let go of.
On top of that it's talking about oil as spice. If anything: it's diesel punk just with an abstraction around the diesel.
The fremen meticulously conserved and saved water to one day turn the planet into a habital environment. That's pretty hopeful, I think. Frank didn't really mature that arc of the story and instead focused, like you said, on religious fanaticism and dogma.
Haven't read the books, but isn't Paul/Leto II taking the golden path ironically an implied long-term hope for humanity? >!Since it's understood humanity will die out otherwise.!<
Leto's golden path involves everyone breaking their dependency on spice (oil) and the only way he's able to achieve this is through a brutal dictatorship.
Possibly if humanity had better self control and wasn't constantly fighting for control of the spice (oil) then the hopeful future could be achieved without needing to remove everyone's freedom and stifle innovation.
Water was a scarce resource and probably doesn't count as renewable. There wasn't much need for solar power since it seems they had better energy sources. Spice was mined and used without consideration for its replenishment.
Herbert was vague about some of the technology used, that I know of. Water collection was kinda the biggest thing.
While the Fremen respected the desert and lived in it, I don't think it really has a comparison to a solarpunk society. Fremen only learned efficient ways to survive, IMHO. By that, I mean they learned ways not to get eaten or get extremely dehydrated. It's a bit more of "living on the edge" with scarce resources.
I only remember the first two books, so that would explain where my knowledge faulters.
Could spice be the renewable resource that I am not giving enough credit? I know a little more, but I risk spoiling things since there is probably a new wave of Dune fans after the last movie.
Due to ecological collapse from terraforming, all the sandworms die off and this makes Leto II take over as a tyrant for 3500 years. Although this was necesarry for golden path. And planet turns back to desert and gets blown up