I was wondering whether it is possible to have a button within Workshop that on click creates and opens a copy of an already existing contour analysis?
Yes that is possible. You need to use the URL option of the button widget using the following pattern:
https://STACK_URL/workspace/contour-app/overview/CONTOUR_ANALYSIS_TO_COPY_RID/copy
You can get the RID of your contour analysis from the details sidebar: