You are using an implicit approximation of the Chen-Gackstatter surface right?
An isosurface doesn’t give you a parametrization, so apart from visualizing the overall form, I think it won’t be much help here.
Here what I’d recommend is to manually build the base mesh, respecting the symmetry and topology of the intended surface, then relax it.
This is a suitable base unit (I realised that instead of 4 valence 6 vertices like I said earlier, there should actually be 2 valence 6 (on top and bottom) but in the middle they combine into a single valence 12 vertex)
When reflected and rotated this gives the full surface, which can then be subdivided and relaxed.
One nice thing about using this particular discretization is that you can separate it into concentric rings, or strips that span across between the boundaries, depending on whether you include that MeshTurn step.