You probably need to build the curve from a mesh-object, since armature and blendShapes only work on meshes. I have an old WIP arrow to be used in a game-project where I have tried something similar. I’m currently not too happy with the results, but it seems that it is possible to use both solutions (rigged and shapekeys).
I used IK-modifier on the top bone to control the arrow on the rig, and had to use 2 blendShapes to control the arrow with the shapeKey solution. The armature-animation can be imported directly in Unity, but for the blendShapes, you’ll have to animate the two shapes syncrouously in Unty (easy to script).
It all depends on how many curves you’ll need to animate, and the quality of the animation and material used.
Alternatively there is procedural animation, you can modify the vertices directly in Unity (calculate world coordinates for each vertice and move them in code), this is very efficient, especially if you want to start and end the animation from any point on the globe, but it requires alotta code and some math
It is possible that someone has a better solution (a rope / elastic simulation) that can be used ?