2PN black-hole binary spin precession works exactly like Kepler’s two-body problem. Not kidding: just define effective potentials and divide the phase space into morphologies. The only things you need are a few timescales to play with.

Michael Kesden, Davide Gerosa, Richard O’Shaughnessy, Emanuele Berti, Ulrich Sperhake.
Physical Review Letters 114 (2015) 081103.
arXiv:1411.0674 [gr-qc].
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Supporting material available here.