I am using these two functions which result in getting the top parent category of a custom taxonomy. These have been tested and work as expected.

However, now I am looking for a way to output the child category instead of the parent.

function get_term_top_most_parent( $term, $taxonomy ) {
    // Start from the current term
    $parent  = get_term( $term, $taxonomy );
    // Climb up the hierarchy until we reach a term with parent = '0'
    while ( $parent->parent != '0' ) {
        $term_id = $parent->parent;
        $parent  = get_term( $term_id, $taxonomy);
    }
    return $parent;
}

function get_top_parents( $taxonomy ) {
    // get terms for current post
    $terms = wp_get_object_terms( get_the_ID(), $taxonomy );
    $top_parent_terms = array();

    foreach ( $terms as $term ) {
        //get top level parent
        $top_parent = get_term_top_most_parent( $term, $taxonomy );
        //check if you have it in your array to only add it once
        if ( !in_array( $top_parent, $top_parent_terms ) ) {
            $top_parent_terms[] = $top_parent;
        }
    }

    // build output (the HTML is up to you)
    foreach ( $top_parent_terms as $term ) {
          //Add every term
          $output = $term->slug;
    }

    return $output;
}