You can’t.
The only cookies you can read with client side JavaScript are those belonging to the host of the HTML document in which the <script> is embedded.
By setting withCredentials you can support cookies in cross-origin requests, but they are handled transparently by the browser and JS has no direct access to them (the XHR spec goes to far as to explicitly ban getAllResponseHeaders from reading cookie related headers). The only way for a cross-origin request to get access to cookies is for the server (which you say you don’t have access to) to copy the data into the body or a different response header).