I'd look into pricing for independent connectivity for your organization in your part of the building. Depending on the options in your part of the country, you might be able to find something affordable. Some 3G/4G modem service might be available without a cap and that can be plugged into your group's router and moved where it is needed.
Alternately, what can be done about getting the school to fix their badly misconfigured filters? What kind of level of service agreement do you have with them?
About the tunnel question, OpenSSH's client can make a SOCKS proxy, look at the -D option in the manual page.
Code:
ssh -l pie -D 4558 outside.example.com
That would set up SOCKS5 tunnel to the outside via outside.example.com. You'd connect to it via port 4558 on localhost. Not all applications support SOCKS however. For some of the others, you might be able to use the utility proxychains.