I have not found an acceptable procedure to do that. But, there is an amazing addon for blender that does wonders transferring shape keys, but it still has a problem with exactly that situation. Perhaps though, if you use the addon to transfer vertex groups first, then transfer the shape keys you might have some luck. also, perhaps scale down the helper geometry of the eyeball and try to transfer the shapekeys using the addon after that. i gave up a while back trying to do it easily, but did find some work arounds. For example, you can use the same procedure i described earlier, and use makeclothes to define the eye as 'Human' then you can use MakeShapes to export the shape keys of the eye itself and bring them in again for other models. the problem with this prodecure is it will only work on an identical model. I spent quite a bit of time saving all the shape keys for the eyes, only to find out when i imported them to a different character, they would not work properly becuase they seemed to be adjusting to the height/shape of the original captured character. Oonce i saw that, i basically stopped trying. I am using UE4 eyes anyway so it did not matter that much to me.
Here is that amazing addon
https://gumroad.com/l/tOKEh
It is free.
Good luck!