Add class to oEmbed video from within post editor?

One idea for the shortcode version, would be to add the custom class via the the native shortcode:

https://wordpress.stackexchange.com/questions/167749/...

then you could add it to your custom wrapper with:

add_filter( 'embed_oembed_html', function ( $html, $url, $attr, $post_ID) {
    return sprintf( 
        '<figure class="video-container %s">%s</figure>', 
        isset( $attr['class'] ) ? esc_attr( $attr['class'] ) : '', 
        $html 
    );
}, 10, 4 );

Then you could e.g. add your own TinyMCE button with a custom form to generate this shortcode?