Serve text/html from wp-json API via WPEngine, headers not being set properly

In case someone encounters this exact problem in the future:

I found the error to be related to the external api not being properly configured to handle a request to a secure endpoint.

Since my site on WPEngine is secure (https), IIS needs to be configured to handle TLS1.2, which is wasn’t (start here for the crumb trail : https://stackoverflow.com/questions/22627977/the-underlying-connection-was-closed-an-unexpected-error-occurred-on-a-send).

The configuration in my question works as designed, though perhaps it could be better designed (you let me know!). The server calling this endpoint wasn’t configured properly.