I think the fundamental problem here is that you're extending from the "TextInput" field but it seems like you don't actually want the user to be able to type into the field, so a text input field might not actually be the best choice for you. The "label" field is just text, but then you'd have to figure out how to signal to the user to click there to begin editing, so I guess that's just a different problem... When I wrote my previous response I wasn't looking at your screenshot with the dropdown ux.
The simplest solution I can think of to use what you've currently got is to override the `widgetCreate_` method (in your subclass of FieldTextInput), call the super method (which returns the html element where it added the text input), and then add the "readonly" attribute to the input.
Maribeth