Besides the Lübeck-Batista surface, there is at least one other doubly periodic minimal surface of genus 3 that could be dubbed a doubly periodic Callahan-Hoffman-Meeks surface. To my knowledge, this surface doesn’t appear in the literature and is currently only numerically established.
Here, the surface maintains the reflectional symmetries from the Costa saddles but loses all straight lines. The Callahan-Hoffman-Meeks surfaces appear to be rotated by 45º compared to the Lübeck-Batista surface. It would be very interesting to know whether these surfaces can be minimally deformed into each other through a continuous rotation of the CHM surfaces.
At the other limit, the surfaces apparently converge to 8-ended Scherk surfaces, stitched together using catenoidal necks.
Resources
Mathematica Notebook
PoVRay Sources
Minimal Surfaces 