Add a leading zero to pagination

wp_link_pages_link filters the entire HTML link (<a href... etc.), which is why what you’ve got isn’t working. There doesn’t appear to be a filter for just the page number, $i, so you’ll need a workaround.

Since the callback function for the wp_link_pages_link filter passes the $i, we could use that to find the number inside the HTML and replace it.

The simple version would be to simply replace $i inside $link:

function wpse_287783_zeroize_page_numbers( $link, $i ) {
    $zeroised = zeroise( $i, 2 );

    $link = str_replace( $i, $zeroised, $link );

    return $link;
}
add_filter( 'wp_link_pages_link', 'wpse_287783_zeroize_page_numbers', 10, 2 );

There are two related problems with this solution:

1. This will also affect the page number in the URL, which we don’t want to do, because this is purely cosmetic.
2. If the URL includes any other numbers this will affect those too. This could include date archives where months or years that have the page number in them.

So what we need to do is only replace the number inside the HTML tag. To do that we can use a regular expression:

function wpse_287783_zeroize_page_numbers( $link, $i ) {
    $zeroised = zeroise( $i, 2 );

    $link = preg_replace( '/>(\D*)(\d*)(\D*)</', '>${1}' . $zeroised . '${3}<', $link );

    return $link;
}
add_filter( 'wp_link_pages_link', 'wpse_287783_zeroize_page_numbers', 10, 2 );

Now someone might be able to provide a better regular expression, since I’m rubbish at them, but what '/>(\D*)(\d*)(\D*)</' does is get characters between > and <, which will be the stuff inside the tag, and return 3 things: Text before any digits, any digits, and text after the digits. The text before and after is so it still only affects digits if link_before, link_after or pagelink from wp_link_pages() have any other text in them.

With preg_replace() we are replacing just the 2nd thing returned, the digits, with the zeroised version of the page number.