There is an open GitHub issue in relation to adding Sign in with Google via a shortcode, which would make your objective easier. Chime in on the below GitHub issue and subscribe for updates:
https://github.com/google/site-kit-wp/issues/10150

There may also be plugins that allow you to insert blocks into shortcodes or into WordPress hooks. Here's one such example: https://wordpress.org/plugins/blocks-to-shortcode/