Eberhard-type theorems for Fisk triangulations?

This is only a partial answer.

First, a sphere does not admit Fisk triangulations. Fisk has proved it by using the 4-color theorem, but there is a simpler proof using Fisk's method of coloring monodromy. Try to color the vertices in three colors. There will be no contradiction when you go around an even vertex; but the colors will be permuted when you go around an odd vertex. For one vertex this will be the $(12)$ transposition, for the other vertex $(23)$ transposition. Thus, the fundamental group of the sphere minus two points will be mapped epimorphically onto $S_3$, which is a contradiction.

The torus does not allow triangulations with vertex degrees $5, 7, 6, \ldots, 6$ (even if you don't require the $5$- and the $7$-vertex to be neighbors). This is a result by Jendrol and Jukovic, reproved recently by a geometric method. There is however a "pseudotriangulation" (in the sense that it is not a simplicial complex) with two vertices, one of degree $3$, the other of degree $9$. If you cut it open along an edge and glue to a $6$-regular torus cut open along an edge, then you get a surface of genus two with vertices of degrees $9$ and $15$. This can be iterated. (Again, the result is a pseudotriangulation.)

But if there is a $(3,9)$ Fisk triangulation of the torus, then there is also a $6$-regular Fisk triangulation of an orientable surface of any genus. Instead of cutting along edges, cut the $(3,9)$-torus and a $6$-regular torus along two-edge paths (for the $(3,9)$-torus take a path between the exceptional vertices. The cut open tori can be glued together so that the odd vertices remain odd and adjacent, and all other vertices keep degree $6$. This gluing will not create a double edge as the previous construction.

EDIT: On the other hand, I am starting to doubt the existence of a $(3,9)$-triangulation of the torus. Maybe one can prove by geometric methods that in the canonical Euclidean cone-metric there are several shortest geodesics between the exceptional points. This would mean, there must be multiple edges in the skeleton of the triangulation.