There’s a topological reason why they can’t all be hexagons meeting 3 per vertex.
The only surfaces which can be tiled with all hexagons in this way are those topologically equivalent to a plane or a genus 1 surface (a torus).
Connected triply periodic minimal surfaces always have genus of at least 3.
I believe it would be possible to make a symmetrical mesh with hexagons and a few appropriately placed octagons for this surface though.
I posted some quad discretizations of the batwing surface here. You could use this as the starting point