Here is the key equation:
driver efficiency * Vbatt * I batt = Vf * If = watts fed to the LED
If you want to drive each die at 350mA, then you are looking at feeding the MC-E about 0.350A * 3.2V ~ a little over 1W per die, or probably around 4.5W total. If you want to send 500mA per die, then you are looking at sending probably about 6.6W total.
With one RCR123A cell, Vbatt will be around 3.7V. If we assume the best case of 100% driver efficiency, then
Ibatt = (4.5W or 6.6W)/3.7V ~ 1.2A or 1.6A.
Regular RCR123A Li-ions are usually around 500 mAh, so that the recommended 2C discharge rate is about 1A. Thus, you probably should use IMR16340 cells for this application.
Whatever driver you consider, it has to be able to accept 1.2A or 1.6A input minimum (in reality, probably more than that since driver efficiency is not 100%) to be able to output 4.5W or 6.6W to the LED.
For a typical efficiency of 80%, your driver needs to accept input currents of about 1.5A or 2A, for 350mA or 500mA drive current per die.
For your application, the 1400mA AMC7135 linear regulator board comes to mind (assuming it fits), running the MC-E in 4P. Efficiency is essentially Vf/Vbatt, or about 3.2V/3.7V~86%.