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: