Add “gift mode” to product

In Woocommerce, in the single product view somewhere before the “Add to cart” button, I want to enable a checkbox to “Mark as a gift”.

Then in the Cart view, I want to append to the title the string “Product title – (Gift)”

When the shop owner gets the order, somewhere there must be a note saying: “This product was marked as a gift”.

I am currently doing this:

add_action('woocommerce_before_add_to_cart_quantity', 'afz_is_this_gift_add_cart', 35);
function afz_is_this_gift_add_cart(){
    ?>
    <p>
    <label class="checkbox">
        <input type="checkbox" class="woocommerce-form__input woocommerce-form__input-checkbox input-checkbox" name="is-gift" id="is-gift" value="Yes">
        <span><?php _e('This is a gift','afz-woocommerce-gift-checkbox'); ?></span>
    </label>
    </p>
    <?php
}

An then before the checkout form:

add_action('woocommerce_before_checkout_form', 'afz_gift_checkout');
function afz_gift_checkout(){
    
   $itsagift = false;
    
   foreach(WC()->cart->get_cart() as $cart_item_key => $cart_item){
      if(isset($cart_item['is-gift'])){
        $itsagift = true;
        break;
      }
   }
    
   if($itsagift == true){
      add_filter('woocommerce_ship_to_different_address_checked', '__return_true');
      add_filter('gettext', 'afz_change_ship_different_address_text');
   }
    
}

But I don’t know how to pass on the ‘It’s a gift” flag into the database so that the shop owner can see what’s going on with this product in the order.

What do you suggest?