Cannot pick category

I have a custom post type “Accommodation”, but I do not want anybody except me to add accommodation listings. Therefore I removed the possibility to add “Accommodation” from the Add Listing menus and from the dashboard.
I have added an “Accommodation” category, called “Cottages”.
But whatever I try, I cannot create any cottages listings now because the category picker does not contain “cottages” or anything else. I keep being told I need to pick a category, but there is no category to pick from.
I bet it has something to do with the fact that I am not allowing anybody except Admin to add “Accommodation” listings …
Everything works fine for other custom types that I allow frontenders to add.