You can set the body of the message using the "body" attribute: see the
documentation.
I suggest you first get to the point where you're able to use "curl" to send a POST request with the necessary headers and body and get the response you expect. If postman won't show you the raw request, then you might want to use tcpdump or wireshark to see exactly what postman is sending.
Once you've done that, transforming it to blackbox_exporter configuration should be straightforward.