So, I was able to find a solution. And funny enough, it was by simplifying instead of complicating. :)
Since the searchform() and global_site_search_form() are supposed to look identical, I added the <?php if ( get_theme_mod( 'sicher_enable_multisite_search' ) == 1 ) : ?> directly into the searchform.php. So when the multisite search is enabled, the action of the form field is set to <?php echo esc_url( trailingslashit( $current_blog->path . global_site_search_get_search_base() ) ) ?> and the name of the search input is set to name="phrase". That's all.
Now my global-site-search-form.php simply loads <?php get_template_part( 'searchform'); ?> and I can use the same function as before, with $menu_item = '<li class="right menusearch searchposition1">' . get_search_form( false ) . '';