I don’t see any option to right click that lets you share a design on the workshop. And when you click the “Browse the Workshop” button in the ship library there also isn’t any obvious way to share designs you made.
How to Share Your Designs
Open your ship library, create a new folder under “My Ships”, and put the ships you want to upload in it.
Right-clicking that folder in the ship library should give an “Upload to Workshop” option.