Sorry for bumping this post but maybe my solution can help someone in the future. Two approaches to this problem are:
1. What I did was create a separate login and registration service and stored all website passwords in my own database. After that when the user clicked the site.pro website builder link on my website control panel (not vpanel) it would automatically load the website builder as the password was already in place. This is one approach (you can look at my implementation here: https://ninth.me/).
Second approach is to create a popup dialog using bootstrap modal for styling or simple javascript dialogs inside of your cpanel.example.com and have the user enter their cpanel password. Then, use a POST request using javascript and send it to a PHP file in example.com which will then establish a connection to site.pro and redirect the user to the website builder.
