You can add your bones to it with a mirror modifier. Just make sure that vertex groups are enabled in the mirror modifier, your bones (+groups) have appropriate .L/.R suffixes, and that the armature modifier is evaluated after the mirror modifier.
I don’t know what you have against using a UV map. But the precise math you’d need to mirror your displacement map (this is not a displacement map btw, a displacment map should be black and white) is going to depend on the precise way in which you’re reading that map, which you haven’t shown. (Generated texture coords? Object coords? Something else?)