How can I script the DuckDuckGo Privacy Essentials browser extension? Click button in window open by browser extension?

The DuckDuckGo Privacy Essentials browser extension (PE) allows you to generate disposable e-mail addresses. I want/need to script that (i.e., generate an address and insert it into a web form). How do you do that?

There are some obvious problems:

  1. PE for Safari doesn’t support these disposable e-mailaddresses.
  2. Firefox isn’t, more or less, scriptable.
  3. The DDG browser is absolutely not scriptable
  4. I don’t use Chrome but I guess that is my only option here, unless there are some API-calls or similar I could use?

Other alternatives?

Can you click on a button in the window an extension opens using a script? (The result is inserted into the clipboard and that is easily accessible)

(This is the interface when you click on the extension. I need to script a click on the Generate Private… button.)

A JavaScript script might be possible. Or a scriptlet

You may be able to do something with UI scripting. But, that could be difficult, and is subject to UI objects being renamed, moved, and so on, so it is fragile.