I didn't see anything listed in the browsers properties either (in the dictionary), and I of course don't spend enough time in the newer IDEs to even hazard a guess as to what would be needed specifically.
Weird that you can copy / paste manually, but not programmatically just because it is a widget. I mean, isn't that what the copy / paste commands are supposed to be for?
And here is the real question, if your in the IDE, and you select text from a pdf in the browser widget, then go to "Edit -> Copy", bring the SE or message box to the front, insert the caret and go to "Edit -> Paste", it sure does copy and paste from the widget.

I have to think there is a way to do it, if you can do it in the IDE, but I am not going to wade through one of the IDEs to find it myself

*Edit - well, I lied, I went and looked

The IDE's copy goes to the menubar stack, then to stack "revidelibrary" for the revIDECopy handler, which apparently checks to see whether what is being copied is an image, text, object, etc.
Unfortunately, I do *not* have the time to determine exactly the flow it goes through to copy from a pdf
