@Zache Did a first implementation of this functionality for the Finnish Wikipedia! Code can be found at https://fi.wikipedia.org/wiki/J%C3%A4rjestelm%C3%A4viesti:Gadget-ProveItCore.js and https://fi.wikipedia.org/wiki/J%C3%A4rjestelm%C3%A4viesti:Gadget-ProveIt.css
I added a button to the interface as you suggested. However, I had trouble making it look like you designed it, because doing so would require adding a third column to the reference table, and colspan="2" to every other field. In the end I thought that complicating the code that much wasn't worth it, so I settled for a floating button with a basic style, which in my opinion looks decent in this context. I also applied this button style to the "Today" button, making the overall look of the gadget a little bit more consistent.
Finally, I did some CSS changes to the List and Add tabs. I hope you like them!