In Chrome (and apparently Firefox), there is a special copy() method that will copy the rendered content to the clipboard. Then you can do whatever you want by pasting it to your preferred text editor.
https://developers.google.com/chrome-developer-tools/docs/commandline-api#copyobject
Console Example:
copy(document.body.innerHTML);
Note: I noticed Chrome reports undefined after the method is run, however, it seems to execute correctly and the right content is in the clipboard.