Exporting animation to Unity problematic, looking like Blender's baking problem

I’m having trouble with exporting a character animation from Blender to Unity.

Everything seems to export out fine except the eyelids animation… it doesn’t close correctly. I’m guessing this may be because of the shrinkwrap and/or action constraints I’m using, but I’m not entirely sure.

At first, when I saw the animation did not export correctly, I thought I could work around it by baking the animation in Blender first (Pose -> Animation -> Bake Action)… but the baked animation in Blender has the same problem.
Tried various methods of baking/exporting, but nothing works.

Strangely enough, if I import in the .blend file itself into Unity, it seems to work. This isn’t an ideal workflow for us though. Can anybody figure out what’s happening?

Problem file is attached. Layer 2 has the geometry being used for shrinkwrap.

Attachments

AnimationBakeProblem.blend (3.38 MB)

Just updating that it turns out it’s possible to apparently bake correctly within Blender if only using the “Only Selected” option.

…But this still doesn’t work as it reverts to being messed up when exporting to .fbx, which is really strange.
And what is it about importing in the .blend file to Unity that causes it to work correctly?

Help would be much appreciated.