How to use the value of a custom Taxonomy ?

Hi,
I have a snippet that uses the value of custom fields to rename the post when it it saved (published or modified). It works very nicely with simple text fields, but how to pull the value of a Taxonomy field ? I cannot manage to pull the value from the related CPT when the saving of the post is happening.

I would like the new title to be based this way :
{current post's value for custom taxonomy"genre"} {value of custom field (text) "objet_designation"} - {title of related post for CPT "designer"} - {current post's value for custom taxonomy"date"}
This way the generated title would look like this (for example) : Chair "Model A" - Starck Philippe - 1981

It is working except for the related post's title that does not show up (stays blank) : Chair "Model A" - - 1981

Here is the snippet in it's actual form, generating a blank space where I want the related post's title to be used :

add_action( 'save_post_objet', 'update_objet_post_title_with_designer_fixed', 10, 3 );
function update_objet_post_title_with_designer_fixed( $post_id, $post, $update ) {
    // Prevent recursion, handle autosaves, revisions, and check for actual update
    if ( defined( 'DOING_AUTOSAVE' ) && DOING_AUTOSAVE || wp_is_post_revision( $post_id ) || !$update ) {
        return;
    }

    // Assume 'genre', 'date', and 'related_designer_id' are correct field identifiers
    $genre_terms = get_the_terms( $post_id, 'genre' );
    $genre = !is_wp_error($genre_terms) && !empty($genre_terms) ? $genre_terms[0]->name : '';

    $date_terms = get_the_terms( $post_id, 'date' );
    $date = !is_wp_error($date_terms) && !empty($date_terms) ? $date_terms[0]->name : '';

    $objet_designation = rwmb_meta( 'objet_designation', '', $post_id );
    $designer_post_id = rwmb_meta( 'related_designer_id', '', $post_id );
    $designer_title = $designer_post_id ? get_the_title($designer_post_id) : '';

    // Construct the new title afresh each time
    $new_title = trim("$genre \"$objet_designation\" - $designer_title - $date");

    // Direct comparison to prevent unnecessary updates
    if (get_the_title($post_id) !== $new_title) {
        // Temporarily unhook this function to avoid infinite loop
        remove_action('save_post_objet', 'update_objet_post_title_with_designer_fixed', 10);

        // Update the post
        wp_update_post([
            'ID' => $post_id,
            'post_title' => $new_title,
            // Optionally update the slug as well
            'post_name' => sanitize_title($new_title),
        ]);

        // Re-hook this function
        add_action('save_post_objet', 'update_objet_post_title_with_designer_fixed', 10, 3);
    }
}