Interesting question. I can find answer using my program, which was made for solving Tammes problem for 13 points. But I need some time for answer.
UPD: I wrote program. Result: 24 is a maximal number of edges. I did in three steps. First, I enumerated planar graphs with 12 vertices with at least 25 edges, at most 5 edges in a vertex and at most hexagonal faces. Total number of suc graphs is 67497.
Second, I eliminated by linear programming by considering values of face angles as variables. My constrains was: 1. angle in triangle is ~1.2310 2. each angle no less than 1.2310 3. sum of angles around vertex is 2*pi 4. opposite angles of rectangle are equal 5. sum of non-opposite angles in rectangle between 3.607 and 3.8213
I solve feasibility of this LP problem (with some tolerance) After this step all graph were eliminated.