I've come up with an alternative for beziers that might be easier to ray-trace, perhaps even though a plain vertex shader. Though there's missing a piece.

I need to find the parametric surface equation from the surface normals I have for edge vertices. I also have to know it's peak and valley so I can constraint the depth of my bounding triangle.

Image explains the overall idea: I build a bounding-volume from a control triangle. Then apply a function to each parametric coordinate of the triangle (s+t+u=1 where s,t,u >= 0) to get the height coordinate for that certain point.

Simply put, it produces a procedurally generated height-map for the triangle's surface. I just need to find a function that generates the height-map so I can make it work.