Woocommerce Order – Rest API – Payment

Firstly, thank you for a great plugin.
I am making use of a third party to receive cash payment for orders placed in Woocommerce, and this is how we want it to work:
The user pays on WC order number, 3rd party sends an HTTP request to WP Rest API to fetch a few details like amount payable, user name, address etc. The HTTP request they send is very specific and has a bunch of fields that I would need to add to the API ( I wish I knew where to edit that). The majority of those fields are not important to us but need to be captured and sent again in the response from the API. One field in particular is a Transaction ID that they generate when a payment process starts.
I added fields in the “Order Custom Fields” option in Woocommerce Orders section, and they show up in the REST API Controller Setting page the ORDERS Category where I’m able to enable them.
The problem is that when I send a GET request with POSTMAN to https://<my host>/index.php/wp-json/wp/v2/cs (cs – edited endpoint name), I do not see anything in the response from the API. I’ve tried the URL without index.php with the same result.
I’ll be most grateful if someone could just point me in the right direction please.