So worst case is that a transistor has to drive 2.4A, they have a max pulse rate of 5A so I guess that's OK (Transistors aren't my strong point).
They have an Hfe of 80 at high currents, so the base current has to be 2400/80 or 30mA, can the ac138 provide that much on a single pin?
One way to measure the current would be to break out the TLC5916 GND pin and measure the current flowing out that. I assume that with all LEDs on there will always be 16 LEDs driven at a given time. Different LED but still 16 so you could use that measurement /16 minus the chips quiescent current to see what each LED gets.
Rob