I asked about plugin to clarify what permission allows to create custom post type. I don’t have access to the “Toolset Types” plugin though. This solution is not suitable in case it uses some widely used capability like ‘manage_options’

URE adds user capabilities for custom post type (CPT) to ‘administrator’ role automatically only when “Users->User Role Editor” page is opening. I added custom filter ure_cpt_editor_roles in order to extend the list of roles to which URE will add capabilities for CPT in addition to default administrator. It will be available with the next update.

URE page should be opened at least 1 time after new CPT was added. Will this update satisfy your needs?