This is why it Hotspot Shield might be able to bypass that setup.
Your browser normally is configured to be connected to a specific network right? Yep! HS bypasses that connection by sending requests to its own VPN and communicating through that network instead of your school's one.
Before attempting anything else, have you tried to take a look at the browsers configuration? Maybe you might be able to alter the proxy settings? Or have you tried to play around with the windows firewall settings?
To find out which ports is HS using simply open cmd and type "netstat -a -b -n", find your process name and take a look at the ports being used. To be honest this is where it gets darker, I doubt you will be able to connect to a different network (unless you do lots of researches on this topic). The most I can advise you is to try binding a socket on that port (after closing HS surely) and then try to set up connections on that specific port.
P.S. It's more likely that your school is only blocking connections via a web browser, have you tried to play some kind of MP game on there? Or any other software that connects to the internet to see if it has access to its online services?