So the computer based solution proposed by Newton and Turing is rejected because the system is chaotic, meaning a general solution will always diverge from reality.
What I don't get is: this should still be good enough Run your solution every month or so, with updated measurements, and you'll have an ongoing "forecast" of conditions.
I'm referencing weather because that's we do. A weather forecast is a prediction of a chaotic system, but of one which changes every day or so. Prediction difficulty is dependent on local conditions and weather type, but we can still make predictions.
A gravitational system of 4 mutually interacting bodies is muuuuuch simpler than weather, and could be predicted far enough in advance to let a civilization adapt and persist!
In the books they basically had the forecast system you talk about. The real problem was their most recent predictions showed their whole planet falling into one of the stars.