Unfortunately, the chip inside the Freedom won't have anywhere near the current drive for the QIII. It's just a tiny 8 pin microprocessor. Five of the pins are connected together to supply about 60mA to the small LED. The QIII would require hundreds of mA.
Perhaps, an external transistor could be added to handle the extra current.
I'm not sure if microprocessor in the Freedom would run on the less than 3V under load from the QIII's single CR123. It normally gets more than 5V from the Photon's two CR2016s.