"I assume I need a DPDT switch to throw the Tortoise..."
No, you need only a SPDT (what we call a form-C switch) with no center off position.
A small AC source and two diodes supplies the power.
The Tortoise® machine comes with two form-C contact sets that allow external switching of track power, pilots lamps or relays to do other stuff.
LED's need resistors in series, or you will release the smoke and they will not be happy.