scswift:
How would I go about selecting an inductor for this purpose?
I've done some sums, and from the datasheet I think the PWM frequency of the TLC5947 should be around 1kHz (based on 4MHz nominal internal clock and 4096 clocks per PWM cycle). If you use a 100uF decoupling capacitor, that will give an impedance of 1.6 ohms to ground. To get a 10-fold reduction in the amplitude of the unwanted signal, the impedance of the inductor should be 16 ohms, which implies an inductance of about 3.3mH. The inductor needs to be rated at the maximum current drawn when all the LEDs are on. The resistance of the inductor needs to be small enough not to cause an unacceptable voltage drop.
You could also use a larger decoupling capacitor (say 1000uF), which would all9ow you to use a smaller inductor (e.g. 330uH for the same 10-fold reduction at 1KHz).
it is probably the high frequency harmonics that are causing most of the problem, so this L-C combination may be rather more effective than the calculation implies.