So all we need to do is to check, what exactly WP is doing during wp_head and mimic that behavior…
In wp-includes/default-filters.php (line 244) you can find:
add_action( 'wp_head', '_wp_render_title_tag', 1 );
add_action( 'wp_head', 'wp_enqueue_scripts', 1 );
add_action( 'wp_head', 'wp_resource_hints', 2 );
add_action( 'wp_head', 'feed_links', 2 );
add_action( 'wp_head', 'feed_links_extra', 3 );
add_action( 'wp_head', 'rsd_link' );
add_action( 'wp_head', 'wlwmanifest_link' );
add_action( 'wp_head', 'adjacent_posts_rel_link_wp_head', 10, 0 );
add_action( 'wp_head', 'locale_stylesheet' );
add_action( 'wp_head', 'noindex', 1 );
add_action( 'wp_head', 'print_emoji_detection_script', 7 );
add_action( 'wp_head', 'wp_print_styles', 8 );
add_action( 'wp_head', 'wp_print_head_scripts', 9 );
add_action( 'wp_head', 'wp_generator' );
add_action( 'wp_head', 'rel_canonical' );
add_action( 'wp_head', 'wp_shortlink_wp_head', 10, 0 );
add_action( 'wp_head', 'wp_custom_css_cb', 101 );
add_action( 'wp_head', 'wp_site_icon', 99 );
One way would be to remove all redundant actions and use wp_head in custom header.
Another solution is calling the functions manually in there. So in your custom header you can do this:
<?php
wp_enqueue_scripts();
wp_print_styles();
wp_print_head_scripts();
?>
and it should be enough.