Yes, if you make a circle packing mesh on your surface, and another circle packing mesh with same combinatorics (on a plane, or on another surface), the mapping between them is conformal.
If your mesh is topologically a disc or a sphere, things are much easier. Also, it depends on how much area distortion is acceptable. Conformal mappings preserve angles, but for doubly curved surfaces this always comes at the cost of some area distortion. This can be reduced by including seams and cone singularities, as shown in your image. Choosing where to put these, and doing the splitting complicates things though.
Anyway, assuming you can treat it as a disc without excessive area distortion, you can flatten it with a Tutte embedding (just using zero length springs guarantees it will be crossing free in the plane). Then if you optimise both the planar one and the one on the surface (while keeping them on these) for tangent incircles, you have a conformal mapping.
There are other ways of doing this with Kangaroo though -
I’ll post some examples tonight. I also have somewhere a goal for the cross ratio energy given in ‘conformal equivalence of triangle meshes’, that might be a way to do this with fewer steps.