How can I create a diffuse map that looks and reacts like a Vertex Weight map? Goal: Create a animation of the gradient pressure map on the top of a seat as the user sits down. 

What I've tried: Dynamic painting works well, but I can't figure out how to get the gradient colors along the borders of the paint similar to the surface type:"weight" setting in dynamic painting. 

It would be perfect if I could figure out how to create a series of images that look like the vertex weight paint mode with surface type:"weight" setting in dynamic painting to use as a diffuse shader.
You have two options:
Image sequence

With the canvas object selected, set the format to image, then set the UV map and output path/format:


Bake the image sequence, then load it into your material with an image node:



Result:

Vertex paint:
You can also bake to vertex colors and use a node setup like this, using an attribute node to get the vertex color: