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But predicting out real world chaotic orbits a few years in advance to the required precision is fairly trivial. You only really need high precision to line up a slingshot for the next slingshot not the slingshot after that. A last minute correction of 1 mile per hour * 1 month ~= 720 miles.

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You're right, the real problem with predicting trajectories through those "keyholes" [1] isn't solving the dynamics equations: our numerical integrators are more than good enough. The problem is we can't measure the initial states with enough precision.

[1] https://en.wikipedia.org/wiki/Gravitational_keyhole

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