modifying the entire engine pipeline to pass the data all the way from the mesh to the shader and output as material nodes is probably not worth the huge effort.
the easiest way would be to bake the curvature into the mesh vertex color or into a separate texture. or making a new UV channel where you store the curvature value as the UV position (you only need one axis in the UVs)
out of the 3 solutions above I’d probably pick the extra UV channel simply because it gives more freedom and versatility. for reference, SpeedTree stores all its tree branch/wind/etc properties as extra UV channels.
using the mesh vertex color has as downside that you lose a vertex color channel for vertex paint if you need that, and baking it to a texture makes it much less versatile as you need to manually pass it to the shader etc.