It seems I have found the way to make WordPress default gallery pagination to work and to have control over URL forming: Pagination with WordPress default gallery? But when gallery call is in post.

When gallery call is in page, it works only if URL is like this: domain.com/pagename/page/2/, domain.com/pagename/page/3/ etc. Trying to have a different URL ends up in stopping it to work.

The part of code handling pagination in functions.php:

// Pagination Setup
$current = (get_query_var('paged')) ? get_query_var( 'paged' ) : 1;
$per_page = 3;
$offset = ($current-1) * $per_page;
$big = 999999999;
$total = sizeof($attachments);
$total_pages = round($total/$per_page);
if( $total_pages < ( $total/$per_page ) ){
    $total_pages = $total_pages+1;
}

// Pagination output
$output .= paginate_links( array(
    'base' => str_replace($big,'%#%',esc_url(get_pagenum_link($big))),
    'format' => '?paged=%#%',
    'current' => $current,
    'total' => $total_pages,
    'prev_text'    => __('&laquo;'),
    'next_text'    => __('&raquo;')
) );

As long as I keep URL like this domain.com/pagename/page/2/ , other ways for ‘base’ also work:

'base' => get_permalink( $post->post_parent ) . '%_%',
'format' => 'page/%#%/',

And like this:

'base' => get_pagenum_link( 1 ) . '%_%',
'format' => 'page/%#%/',

But when I try some other URL scheme:

'base' => get_permalink( $post->post_parent ) . '%_%',
'format' => 'paging-%#%',

pagination stops working.

The filter with rewrite tag and rule:

add_filter('init', 'post_gallery_add_rewrite_tag_rule_2022');
function post_gallery_add_rewrite_tag_rule_2022() {
    add_rewrite_tag('%current%','(.+)');
    add_rewrite_rule('(.+)/paging-/?([0-9]{1,})/?$', 'index.php?name=$matches[1]&paged=$matches[2]', 'top');
}

It seems like add_rewrite_rule is not functioning when gallery call is in page? Or what? Any ideas?