I'm adding one more post, for anyone that runs into this thread trying to achieve something similar.
It turns out that the solution of adding hreflang tags to the body isn't respected by Google search, so there's no point. Testing with the w3 validator https://validator.w3.org returns errors "Error: A link element must not appear as a descendant of a body element unless the link element has an itemprop attribute or has a rel attribute whose value contains dns-prefetch, modulepreload, pingback, preconnect, prefetch, preload, prerender, or stylesheet."