Allow Editor to view/modify a custom post type

I suspect the problem is not when you register the post type but when you create the menu pages.

When you call add_submenu_page() make sure that the $capability argument is correct. In your case you want ‘edit_posts’:

$capability = 'edit_posts';
add_submenu_page( $parent_slug, $page_title, $menu_title, $capability, $menu_slug );