That explains a lot, thanks!

I was considering eventually swapping out my 1024 BBD for a 4096 one in the hope that the clock would be ultrasonic (at least when having an audible delay length comparable to the 1024 at slower settings), and therefore cut off by my breakout box's internal filter.

At any rate, if you're recording into a DAW, then the cheapest and easiest way to cut out the clock while preserving the other high frequency goodness of the delay signal is to use a series of software notch filters with realtime display, such as Reaper's ReaEQ. The clock signal seems to have a few harmonics, but a handful of very thin notch filters can take them all out. This works well as long as you don't change the speed of the BBD in any given recording. By comparison, whenever I try lowpass filtering the BBD, it takes out too much of the actual effect along with the clock signal.

Hope that helps,
Zoë.