In Custom Order Number Lite/PRO plugin, in the Database, we do not change the original order id created by WooCommerce because that order id is used for many core functions and also by multiple plugins and themes.
So what you will have to do is, you will have to contact to support team of the third-party plugin and ask them to make the necessary changes. They will have to make the changes where Order ID is been fetched and have to use the below code for Custom Order Number instead of the code used for Order ID.
Here is the code they will need.
$custom_order_number = $order->get_order_number(); //where $order is an order object