How to use apply_filter to wp_link_pages in functions file

Hello!
At the link I shared, I’m still using the twenty twenty theme (with a child theme)

What I need is to replicate what I did for paginated posts (see the link), with the twenty twenty two theme.
Basically I split my posts in two, to show the footnotes on a next page; and I use the pagination to get “see footnotes” and “back to article” buttons instead of: “Pages: 1, 2”
Of course, this is only meant for posts like this, split in two pages.

Since the twentytwentytwo theme doesn’t allow to change the PHP file, I need to use the apply_filter() for wp_link_pages(), and I have no idea how to do that.

So far, I managed to do the following with add_filter() instead:
(in a child theme for twenty twenty two)

add_filter( 'wp_link_pages_args', 'my_function');
function my_function(){
  $args = array(
                            'before'      => '<div>',
                            'after'       => '</div>',
                            'link_before' => '<span class="page-number">',
                            'link_after'  => '</span>',
                            'next_or_number'    => 'next',
                            'separator'         => ' | ',
                            'nextpagelink'      => __( 'See the footnotes', '' ),
                            'previouspagelink'  => __( 'Back to article', '' ),
                            'echo'              => 1

  );
  return $args;
}

It works, but since I’m not using apply_filter() as I should, I see the pagination also above the page (on the upper left corner, inside the body)
I guess it’s because I’m not using the hook I need to use and because I’m not defining the $output.

I have no idea how to do it.

Thank you so much in advanced!

The page I need help with: [log in to see the link]