When a service call is made via the API Connector and the server response includes the Set-Cookie header, this cookie should be saved by the API connector and reused in subsequent calls to the same service call domain by using the Cookie http request header with the value returned from the first API Set-Cookie header. This is part of the http specification, is implemented in all browsers and REST clients, and is a necessary requirement to be able to use many different REST service APIs with Bubble. See https://developer.mozilla.org/en-US/docs/Web/HTTP/Headers/Set-Cookie for a full description.
Essentially, the API Connector plugin and Bubble are unusable for us without this most basic HTTP client functionality.