I still cannot find a symbol for the Royal Navy Medical Service, but have found the one for the Royal Naval Medical Club. https://www.facebook.com/pages/Royal-Navy-Medical-Club/317270361779099
If posters scroll down many images of the symbol can be seen. It contains an anchor, which Arthur saw on the Duchess' badge, as well as the serpent.
In my opinion the cross seen in the Duchess' badge reflects the Red Cross which medical personnel wear to identify their role.