I did some internet research, and I think I understand the root cause of the flickering light.
According to this post from Eric, these switches support an inductive load of up to 400 watts.
Inductive loads are loads powered by magnetic fields, such as fans or motors. My Google-fu has taught me that inductive loads do not play nicely with relays. When a relay cuts the power to an inductive load, the magnetic field collapses. This collapse creates a quick voltage spike that can cause an arc inside the relay. The arc leads to premature failure of the relay.
To combat this, relays for inductive loads typically use whats called a snubber (essentially a resistor and capacitor across the relay). A snubber’s primary function is to limit the voltage spike in order to prevent the electrical arc. A side effect of the snubber is that a small amount of current is allowed to pass through the switch even when it is off.
For an inductive or resistive (incandescent light) load, this small amount of current isnt enough to do anything, but for a capacitive (LED/CFL) load, this can have unintended effects. Instead of passing through the bulb harmlessly, it slowly charges the capacitor inside the bulb. When the voltage in the capacitor gets high enough, the light turns on. The light drains the capacitor pretty much immediately (causing light to flash), the light goes out, and the capacitor starts charging up again.
If this is the case, it looks like the only real solution is to somehow change the load. It might be changing one or more bulbs, or installing a bypass.
I replaced my 2 CFLs with 2x Fiet Electric CEOM100/930/4 bulbs from Costco. These bulbs are considerably brighter than the old bulbs (which is fine by me since they are in the kitchen), and it solved my problem.
Disclaimer: I am not an electrician, so I cannot say for certain that any of the above information is correct (except that changing the bulbs out solved my problem).