You want some nice 3D features even if some are achievable most of the time effort does not worth it.
Attached image shows what is actually happening when we do head turn control using spine features (top), what you want is bot.
If you constrain eyes on one controller, and use with different values for them you will always get your current result.
You may need 2 controllers, one for left right up down, other for depth. But I never actually was able to made it myself however hard I tried.
Also with 4.x Spine version they have made some improvements on constraints (separate X, Y axis) that can come handy
But apart from curiosity and completing the challenge I would really not suggest going so far, especially for this kind of character.
In most cases task is to deliver the emotional experience of the character and not showcase your skills or spine features.
My chaotic video with more info.
https://youtu.be/aX3a1ar6CXg
Some related tips
Spine: Tips: 32 eye limit
Spine: Tips: oval eye
Spine: Tips: 2 eyes