Maybe it helps if you open the Web Developer tools in a separate window instead of as a panel at the bottom of the Firefox window.

That does work. It's not ideal though. I should also add, if I enable Firebug it has the same key shortcuts and F10 works there.

This may help;

Customizable Shortcuts Allows to customize Firefox shortcuts.

Customizable Shortcuts does not work. It doesn't let you set the debugging keys or the F10 menu key, which I believe is a general Windows key shortcut.