Change custom post taxonomy values from front-end

That would be achieved using the wp_set_post_terms function.

https://codex.wordpress.org/Function_Reference/wp_set_post_terms

You could run an ajax request on the change event of the dropdown to either wp-admin/admin-ajax.php or a custom JSON API endpoint (preferred but more work) with a payload like this

{
    new_status: 123
    post_id: 321
}

Ensure 100% that the user that is logged in has authorization to modify the post

$post = get_post( filter_var( $_POST['post_id'], FILTER_SANITIZE_NUMBER_INT ) );
if( /*Test User Auth*/ ){
    wp_set_post_terms( $post->ID, [ 
        filter_var( $_POST['new_status'], FILTER_SANITIZE_NUMBER_INT ) 
    ], 'status' );
}