If the discrete asymptotic network is what you are ultimately after, I imagine it would be better to generate this more directly, instead of trying to make a triangulated mesh or continuous surface then numerically trace curves across that (which does seem prone to inaccuracies).
With the discrete minimal isothermic surfaces from Koebe polyhedra as linked in my last post, if you connect the incircle centres to the corners of the quad grid (which follows the principal curvatures), then you get appears to be the discrete asymptotic grid with planar vertex stars you are after:
I suspect there’s also a simpler way possible where you would optimise directly for the planar (or spherical) vertex stars without going via the Koebe polyhedron.