Hi Phil,
CEF does not currently support the Webkit Inspector. Adding support for it would be relatively simple -- a matter of reviewing the test_shell implementation and bringing over the relevant bits. My only minor concern with adding the Inspector in the current version of CEF is that it uses separate CSS/JS/HTML files (see the resources\inspector directory). This type of distribution complexity is something that I feel is generally better targeted for CEF2 which will be based on the complete Chromium distribution.
That said, feel free to create an entry for this on the CEF issue tracker. If you have the ability/inclination, patches to add the support are also welcome.
Regards,
Marshall