Regarding boost converter and current sink IC operation
The TLC5940 doesn't vary the voltage.
It varies the current.  It acts like a potentiometer that automatically adjusts its resistance to keep the current constant.
The TLC5940 also uses pulse width modulation to vary the brightness of the LEDs.
You set the maximum current to the LEDs with a single reference resistor, then you can vary the brightness from off (duty cycle 0 percent) to full on (100 percent.)
The only concern you have with the boost converter is that it (and the lithium cell) can provide enough current for all of your LEDs.