I believe the stumbling block is the original header data of the client. The IP address of the client is not contained in a header, but determined by the TCP/IP layer of the socket that the client connected to. That's what would be in the logs. So, it can't be done quite so simply.
The client's IP can be retrieved by your server via:
$clientIP = $_SERVER['REMOTE_ADDR'];
If you were able to come up with a mechanism to let the other server know that this was a request on behalf of a client, you could certainly send them this data. As you point out, perhaps try setting the
HTTP_X_FORWARDED_FOR header in your request to that server.
Best of luck!