Finding isothermic minimal surfaces like this without having the boundary defined first can be a bit tricky.
If I recall, for the discrete Costa I posted years back I used some bending or co-circular force for the boundary.
Another way to go about this would be to relax a circle packing quad mesh on* the sphere and get the Koebe polyhedron, then rearrange the edges to make the isothermic minimal surface, like in Stefan’s thesis.
*note though that the edges need to lie tangent to the sphere, not the points on the sphere.
It gets a bit complicated around the singularities, but there might be a way to model only 1/4 or 1/8 of the Costa surface to avoid this, then get the rest by symmetry.
Another approach I’ve sometimes thought about is that maybe there is a way to relax the quad mesh towards an isothermic minimal one without explicitly building the Koebe polyhedron or fixing the boundary.
With the tangent incircle quad energy, the grid lines already tend to follow the principal curvature directions, and principal curvatures being equal and opposite at each point is another way of characterising a minimal surface, so maybe a goal to just enforce this angle condition directly…
As for modelling the base quad mesh - I don’t have any really easy way, just practise. If you have or can plot the principal curvatures, you can make some big faces connecting up the singularities.