To be honest, other than reading through the Knowledge Base article and that ZAM thread I posted earlier, I don't have many other ideas for you. I don't have a whole lot of experience in dealing with school networks.. and you are limited in the number of ways you can troubleshoot problems with that type of network. If you can get in touch with your IT people they might have some ideas, but I highly doubt it.
The problem is obviously something that is more or less specific to FFXIV.. but you should still run a long series of tests at pingtest.net to see how stable your connection is.