I believe you're underestimating the state-of-the-art. See [1], for example, which claims a method for saying that a sufficiently good numerical solution implies an exact periodic solution close by. (I doubt that the author of the original article did anything like this.)

[1] http://ww2.ii.uj.edu.pl/~kapela/papers/nosym.pdf

Well, that's the question - is his 416-planet system chaotic? Do perturbtions get amplified or do they get pushed back to the stable configuration?

Of course, when you're talking about a civilization building 416 equal-mass planets equally spaced in 8 counter-rotating orbits, worrying about how stable they are is kind of silly; as long as they're "mostly" stable the maintenance of monitoring and tweaking their orbits for a billion years would presumably both be feasible and cheap compared to the initial construction.

Exactly