I turned off the water to each toilet (and then all of them at the same time), and the knocking still persists.
The two valves (one shut off, one not) go to the outside spigots (front and back). The knocking sound occurs regardless of their on/off setting.
Lastly, another video of the knocking. Sorry for the initial camera movement. The knocking comes on very suddenly at the 19 second mark, grows in intensity and volume, and then stops after about 4 seconds. It starts and stops in spurts like this, sometimes once or twice, sometimes many times over a 30 minute period.
If it is the pressure reducing valve, any thoughts on how much that costs to replace?