Since version 1.15 it is possible to activate the development tools. This works only in the editor, debug and development build.
All you have to do is check the "Allow Dev Tools" checkbox in the HTMLMenu widget settings and press the familiar CTLR + SHIFT + i key combination.
It is important that the browser widget has the focus, so if necessary click on the browser before pressing the keys. Then a new window with development tools should open.