Instead of replacing browser's contextmenu with yours, you could add entries to it, so you won't lose "Inspect" command.
To do that, you can use HTML5 <menu>.
You can read a tutorial here: http://davidwalsh.name/html5-context-menu
<section contextmenu="mymenu">
<!-- Content -->
</section>
<menu type="context" id="mymenu">
<menuitem label="Refresh Post" onclick="window.location.reload();" icon="/images/refresh-icon.png"></menuitem>
<menuitem label="Skip to Comments" onclick="window.location='#comments';" icon="/images/comment_icon.gif"></menuitem>
<menu label="Share on..." icon="/images/share_icon.gif">
<menuitem label="Twitter" icon="/images/twitter_icon.gif" onclick="goTo('//twitter.com/intent/tweet?text=' + document.title + ': ' + window.location.href);"></menuitem>
<menuitem label="Facebook" icon="/images/facebook_icon16x16.gif" onclick="goTo('//facebook.com/sharer/sharer.php?u=' + window.location.href);"></menuitem>
</menu>
</menu>
But be aware that it is not widely supported.