Been modeling shadow bands using Matrix for over 10 yrs. If u have the model, because u made the engagement ring, no problem.
I take three views of the ring (top, end, and through finger), scale to finger rail after trimming away the unwanted parts of the images. Then just model, just takes a little time. Have a video somewhere I can send or post on making shadow bands in matrix. You can import image using "View" > "picture frame". These are surfaces that can be trimmed.
Model using curve from 2 views, etc.
On the subject of a "cheap" 3d scanner that will give good enough results to give u a mesh file to model with, the short answer is, "there is no such animal". If u got 20k to 30k then there are some out there i think that will do what u want.
Dance like nobody's watching!