A shape key is nothing but a curve.
by default, the exporter does not pack custom curves.
You can try using a custom curve and my addon to export an animation file with the custom curve, but depending on the amout of morphs this can be rather complex.
You also need to set up something that reads the curve value and applies it to the morph within the animation BP or the character BP.
I would suggest you create a custom interface function, so you can call it from a variety of objects.
As far as the code, you can literally call set morph target on the mesh, but you have to pull the curve out of the from a reference to the animation blueprint. This requires 1 cast. At begin play, and storing the reference.
Mesh > get anim instance > cast to animation BP > promote to variable.
this way should preserve fast path and avoid code inside the animbp.
alternatively, you could dirty up the animation bp with a ton of Set Morph Target nodes. But the Get Curve Value to Set Morph may actually disable fast path - of that even matters to you.
so working on the character end with reusable code in a component may be preferable.