I was also having a conflict with a custom post type when CF7 was activated.
I developed a new post type & taxonomy called "slidepost" and "slideshow." I did this on my test system and it worked fine, but when I installed the theme on my production site, which has CF7 installed, my menu didn't show up. Deactivating and reactivating plugins one-by-one narrowed the conflict down to CF7.
I was able to correct the problem by changing my menu position to 8. Of course that put it right under the posts on the menu, somewhere I don't really want it, so I was happy to see this:
You can always use this plugin to change the order of your menu items in the admin menu: http://wordpress.org/extend/plugins/admin-menu-editor/
Thanks for sharing the link to the Admin Menu Editor plugin. It works great!
I'm not sure if the problem is with CF7 or my own code. But this solution worked for me.