Usage of custom post types in a custom admin menu

Register the post type without a UI ('show_ui'=>false), then you will have to to create a form on your submenu page(s) and one or more handlers for those forms. wp_insert_post/wp_update_post can handle the post insertion if you give them the right information. Based on the information available, that is the best I’ve got.

For a custom post type, you’d never go to “Posts >> Add New”, though. You’d go to “CPT-Name >> Add New”. They are distinct menus.