Looks like it's time for a redesign. Would a PCA9685 be appropriate? It's an 16-channel PWM driver with totem pole outputs (ie, can drive high or low) and I2C interface.... I know they're popular and there are Arduino libraries for them.
Oh no! I can't use pnp transistors because my LEDs are common anode.
You are not using the chip in the intended way. The chip is intended to sink LED current, not drive the base of a transistor.
You must use a pull up resistor from the output of the TLC5940 to the 5V supply. Then it will work.However because you are inverting the switching you will never be able to turn the LEDs fully off, they will always be slightly on.
+1. It can supply 60mA. You don't need a transistor.
That 220 ohm resistor in the schematic is actually supposed to be 22 ohms.
You didn't think it was important to tell us?