How do I rewrite a single category link to point to a custom page?

Your goal seems to be to change the link which is output when listing the categories rather than making the existing link point to a different page as my last answer assumed. So I am going to try a different answer with a different solution.

Using the filter term_link in the function get_term_link() you can change the link which is generated for a specific category.

function wpse_317747_filter_term_link( $termlink, $term, $taxonomy ) {
    if ( "category" !== $taxonomy || "dogs" !== $term->slug ) {
        return $termlink;
    }
    return get_permalink( $target_page_id );
}
add_filter( "term_link", "wpse_317747_filter_term_link", 10, 3 );

This changes the generated link if working on category taxonomy and the current term slug is dogs. Just set $target_page_id to correspond to the page you want the link to point to.