If your teachers have steam access, and this is something your school is allowing, why not just have the teacher log into the machines?
Alternately, you should be able to contact the Sydney office and explain the situation and have them temporarily give you access to the Steam servers if it's allowed by your school. You'd probably want your principle to contact them in this case rather than calling yourself.
Also, if you can play the game over direct peer-to-peer network, you don't really need to be online. Just have the teacher download the game to one machine, then copy over the local network (this will save the school some bandwidth too). Not absolutely positive, but I've never had any issues with copied games running as long as my Steam account has the game. You probably want to make sure each offline Steam copy is running a unique account though. Steam obviously would try to put in checks to prevent copies of one account playing the same game together.
Worst case scenario, you could possibly rig a cell phone as a local access point, though where I'm from it typically costs extra. You don't want to download that way, or it will cost a ton for data, but connecting to the Steam servers shouldn't be too bad.