With the personal project I've been working on, early on I ran into an issue with blendshape animations not properly exporting with an FBX. My pipeline requires the animation FBX files to only contain the baked joint information and then be targeted to the mesh in Unity. This retargeting breaks the blendshape animations.
To get around this, I added solo joints for each face shape that has blendshapes and set up driven keys to animate the translation, rotation and scale axis when a blendshape is used. For example, the blink blendshape's weight is 1, the eye solo joint moves to 1 on the X axis. Those joints are baked with the rest of the animation. My script in Unity watches those solo joints and adjusts the blendshapes based on those values.
This method also makes it easy to change up the appearance of the eyes later on as well. But I think we're pretty happy with what it is at the moment!
Model by https://twitter.com/Detcelfer81, animation by Ben Hopper and rigged by me.