Your example path was entirely straight lines; does your tool always work with straight lines? You can draw a matching path in Skia by building an SkPath consisting of rectangles at each of the line points (where the rectangle is the size of the brush), and then adding trapezoids between each rectangle edge and the next rectangle's matching edge. This approach seems like it would work for any polygonal brush shape, but doesn't extrapolate to curves.
It's a bit verbose, but this fiddle seems to match your example picture. I kept things simple and just used moveTo and lineTo instead of leveraging the more advanced SkPath APIs:
Using the logic from the variable-width stroker also seems like a good option if you want to support curves; feel free to adapt it to your app as necessary.
Unfortunately, we don't have SkSL support for path effects. SkSL can be used to write custom color filters, or custom shaders, but you can't generate a custom path with SkSL at this time. SkMesh does support applying a shader to vertex data, but it only takes a mesh as input, and is limited in various other ways (e.g. there isn't any CPU rendering support at all).