@Nathan Navarro I attempted some more testing tonight, and ran into a potential issue that effects my testing (but isn't exactly relevant to my core issue).
I tried the alternate program gastric suggested (snoize MIDI Monitor) but not what adamaarts suggested (MIDI Ox does not run on Mac, I didn't realize that). Plugging the Beat Buddy into those programs I still could not see the MIDI clock or Start/Stop commands that the Beat Buddy is obviously sending based on the reactions of my other pedals. However, I did see the MIDI Notes that the Beat Buddy puts out when it is playing.
My hypothesis is that my USB-MIDI interface---an M-Audio UNO---may, intentionally or unintentionally, filter some things out. That's just a hypothesis---I didn't test that yet, because I wasn't sure how (more on this in a bit).
To be clear, the UNO is not normally in my MIDI chain, so this doesn't change anything I'm reporting about the interaction of the Beat Buddy and Hub. And I've seen the UNO record PC, CC, and Note On/Off messages before, so I am still skeptical that the Beat Buddy is passing one of those along. But on three different programs (and two devices, laptop and iPad) using the UNO as my interface, I still can't see MIDI clock, etc.
It didn't occur to me until later that I can use my Zivix PUC as an interface, so I will try that later to see if it improves the monitoring situation.