Custom Post Type UI Members plugins – issues with capabilities

I have Custom Post Type UI and Members plugin installed. I have created a new role which needs to have access to just one post-type (projects) and not see any others. The issue I am having is users in the new role cannot make new posts with any post type whose capability type has been changed from its default; “post”. The error message is “You are not allowed to edit this post.”.

What I’ve tried
Adding new capabilities for the new role using the Members plugin. The Custom Post Type capability type is “project”, therefore I added the following capabilities to the role:

delete_others_projects
delete_private_projects
delete_projects
delete_published_post_projects
delete_published_project_projects
delete_published_projects
edit_others_projects
edit_private_project_projects
edit_private_projects
edit_projects
edit_published_post_projects
edit_published_project_projects
edit_published_projects
publish_projects
read_private_projects

Is there any capability that I’ve missed? I have basically checked every capability visible in the capabilities list, and no luck.

Also note that the Custom Post Type UI doesn’t modify the functions.php file and therefore adding capabilities via code is not possible AFAIK.

Any advice is much appreciated.