OK, so I've heard back from two experts in the field. One from England and one from the US Army research center in Natick, Mass. OP if you want their contact information I can email it to you, but I'm not posting names on here - I at least have to protect upstanding individuals from our little dysfunctional community here.
Anyway, unfortunately for you and this stupid rule, both actually say that a t-shirt is not enough to significantly raise core temperature over running shirtless, even inhot humid conditions. It does however greatly affect the COMFORT of the individual from their work. The athlete/subject FEELS cooler but the core temp is not the same. In fact, one did mention that depending on the severity of the sunshine, a white shirt can actually contribute to lower body temperature over running shirtless. Now, interestingly, both mentioned that some of the new wicking clothing acctually does work to slightly, but significanly, maintain a lower core temperature while exerting in hot environments. They work better than cotton shirts, and they work better than being shirtless (even though the shirtless stuff is not published yet).
So, I guess for your team, stress that they wear tank tops, or running singlets, or have your school pony up for the tech shirts. Kinda sad that not every school will, but that's the current world we live in.
Sorry I couldn't be more of a help to your plight. I still think it's a ridiculous rule.