2 WP websites same content different themes

What I would do is to use switch_to_blog function just before posts queries are run on the database, and then restore the current blog just after the query.

I assume that the blog where your content is is the main one (1). Just put the following lines in your second blog theme’s function.php :

add_action('pre_get_posts', 'user16975_get_content_from_other_blog');
function user16975_get_content_from_other_blog(){
    switch_to_blog(1);
}

add_filter('the_posts', 'user16975_restore_initial_blog');
function user16975_restore_initial_blog($posts){
    restore_current_blog();
    return $posts;
}

This should work for both back- and front-ends.
Be careful to test this fully before using, because it could have unintended behaviors. For example, all the settings used for the second blog are taken from the second blog database, so they might be different from the main blog.