If it is not important which server responds to the client, you can have the form action pointing to your script. There you can do what you need to, and then post a request to the other server's script with LWP, and then send a response back to the client.

If GET is used, you can issue a redirect from your scripts' response, and no more worries. If POST is used, then the redirect may cause the browser to ask users' permission to repost the form, and this may or may not be of a concern. If it is of concern, then you need to do some sort of work-around to accomplish that, and the AJAX solution could get you going.