Hi,
I wish I could read at least something on your linked website, then I can probably help you.
Anyways, I think curl will do it for you,.. you just have to look at the source of the form. Look for the name value of the submit button and then the form processor document.. allow your curl to follow external link wherever the form processing is located.
Here is a demo of a similar script that I wrote before. This script is running in three servers. There are two remotely located servers where the form query is submitted and script will wait for the servers response and then print those responses on the local server (third server).
So, I don't see any complexities on what you are trying to achieved..