Just a question -- are displacement maps keyframable?

If so he can create a Normal Map between the two objects, use

https://66.70.170.53/Ryan/heightmap/heightmap.html

that tool to turn it into a displacement map, then animate from off to on (or from 0 displacement to whatever displacement he wishes)

It won't use Blendshape, but should get the job done in that situation.

(while at the same time keyframing between the two shaders)