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This is a beautiful idea, would love to read a novel based on this premise.

The sort of forced nomadism from "only" having ~1000 years before a location becomes uninhabitable would make for some interesting dynamics. Real estate at the leading edge would be very valuable, potentially with some kind of homesteading dynamic for claiming the land (as nobody has lived there for thousands of years). Also value of land would depreciate over time because it would have an 'expiration date'.

Eventually, the surface-dwellers start to migrate into a region where a previous civilization went underground, and there is dramatic conflict as the nomads try to claim territory that the diggers never abandoned.

Also, the polar territories would only have to worry about precession, and their land might have a longer expiration date.

Start with a tribe nestled in an Eastern valley, with just enough shade and a lake, isolated from hostile neighbors to the North and South.

But soon the Bright comes, melting the glacier which holds the lake back from the Dark. They can't cross, it's now full of chunks of ice, and they have to get through hostile neighboring lands.

Call it Sunrise.

As for fiction w.r.t celestial dynamics, here are two of my favourites that come to mind: Cheela Series and Three Body Problem.

I think its doubtful a society like that would adopt a capitalist land rights system.

First opportunity is always something to capitalize on.

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