Everything turned out to be much simpler.
In system / library / cart / cart.php, in the add () function, we find the code:
if (!$query->row['total']) { $this->db->query("INSERT " . DB_PREFIX . "cart SET api_id = '" . (isset($this->session->data['api_id']) ? (int)$this->session->data['api_id'] : 0) . "', customer_id = '" . (int)$this->customer->getId() . "', session_id = '" . $this->db->escape($this->session->getId()) . "', product_id = '" . (int)$product_id . "', recurring_id = '" . (int)$recurring_id . "', `option` = '" . $this->db->escape(json_encode($option)) . "', quantity = '" . (int)$quantity . "', date_added = NOW()"); } else { $this->db->query("UPDATE " . DB_PREFIX . "cart SET quantity = (quantity + " . (int)$quantity . ") WHERE api_id = '" . (isset($this->session->data['api_id']) ? (int)$this->session->data['api_id'] : 0) . "' AND customer_id = '" . (int)$this->customer->getId() . "' AND session_id = '" . $this->db->escape($this->session->getId()) . "' AND product_id = '" . (int)$product_id . "' AND recurring_id = '" . (int)$recurring_id . "' AND `option` = '" . $this->db->escape(json_encode($option)) . "'"); }
and change it to this:
$this->db->query("INSERT " . DB_PREFIX . "cart SET api_id = '" . (isset($this->session->data['api_id']) ? (int)$this->session->data['api_id'] : 0) . "', customer_id = '" . (int)$this->customer->getId() . "', session_id = '" . $this->db->escape($this->session->getId()) . "', product_id = '" . (int)$product_id . "', recurring_id = '" . (int)$recurring_id . "', `option` = '" . $this->db->escape(json_encode($option)) . "', quantity = '" . (int)$quantity . "', date_added = NOW()");
After that, everything works at all levels as needed - basket, admin, letters to the admin and buyer.