I don't know a great deal about that, as I don't have a computer with an internal mic myself. I can't find anything specifically about the compressors that may be hard-wired in to one, although it does make sense that they would.
I don't think it should cause too much of a problem, assuming Cassidy does her homework and tests, tests, tests what she's trying to do (and I will happily walk her through it and let her know when she's getting what look like reasonable results - i.e. recording at a reasonable volume, and with the tick a reasonable amount louder than the tone). A compressor may make the ticks quieter, or get quieter over time (depending on the settings) but unless everything is mega-loud to start with, I can't see it actually reducing the amplitude of the ticks to the point where they can't be discerned clearly on the waveform, even when the tone is also present.
But, as I say, that's the kind of thing that we can only really get to grips with once recordings start to be made. At the moment everything's too theoretical to make anything too concrete. I don't predict a hard-wired compressor causing any problems to the test I've proposed, but you can never know for sure unless you're actually recording stuff and seeing what's happening.
Oh hang on, I think I might have misunderstood what you're saying. Are you offering the compressor as a potential cause of the drop in volume of what Cassidy describes as the "hum"? So the hum is ongoing, then she taps the mic or somewhere near that's quite loud in comparison to the hum. The loudness of the tapping causes the compressor to kick in and reduce the volume of everything, and Cassidy is tapping at a fast enough rate relative to the release on the compressor that the volume never increases again. Is that what you're proposing?
That's plausible, although it wouldn't explain the vocals disappearing on "Lovely Day". It would certainly be eliminated by the initial testing I'm proposing Cassidy does, though. I didn't say it explicitly when I was typing out the proposal above, but you can see that I've done it in the screenshots I posted - I'd suggest having the metronome at 60BPM, or 1 tick per second. That's close together enough to be considered regular, but I'd be very surprised if it were not far apart enough for any hard-wired compressor to release between each tick.