Yankelle,
“As for the thermos/cooler idea, unless you have something in there maintaining the interior cool, wouldn’t the heat only amplify inside? I think that it might make it worse, not better.”
Even without something cold inside, the insulation of the cooler will help slow the rate of heating inside the cooler as the car heats up, and could make a big difference. Adding a cold pack would keep things even colder, but moisture from condensation may become a problem. If the tefillin are in a plastic case or bag though, they should be protected from moisture, but I would not use ice in a cooler with tefillin.
“Maybe the solution is to keep the entire car cooler by covering the windshield with one of those metallic sheets?”
That will definitely help, but unless the car is in a garage, some sunlight will come in through the windows and the inside of the car will still become much hotter than the outside air.