How do I calculate the required value for a pull-down resistor in a PWM circuit?
In the circuit as you have drawn, R1 does nothing. The 74HC595 has totem-pole outputs.
As such, the output is actively driven either high or low. When QB is high, the output will source however much current is required to drive QB to ~5V. When QB is low, it will sink however much current is required to drive QB to ~0V. 

I think you are not understanding how PWM actually works.
The whole idea with PWM is that you turn something on and off fast enough that the square-wave PWM signal is basically integrated by the system it is being fed into. 

In the case of lighting, the integrator is your eyes.

As such, the proper value for R1 is infinite, and you need to get rid of C2 entirely.