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I want to use "Origin to Geometry" on multiple objects but only move the objects' origins on one axis. How do I do this?

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Bounding box center

IIRC center to geometry is also the same as bounding box center.

Test script, finds the bounding box center cog and only translates the z component.

import bpy
from mathutils import Matrix, Vector
context = bpy.context
ob = context.object
me = ob.data

bbox = [Vector(b) for b in ob.bound_box]
cog = sum(bbox, Vector()) / 8

me.transform(Matrix.Translation((0, 0, -cog.z)))

Changing last line to

me.transform(Matrix.Translation(-cog))

will move origin to center of bounding box.

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