I actually figured out how to disable the beep without destroying it or having to cut the case up.

I gently pulled the motherboard out through the battery opening until the sound box was exposed in the display window, but not far enough that the exposed copper wires to the temperature gauge would break. I poked the sound box with a bobby pin until I could see the beeper's coated wires. This took a bit of delicate jiggling and shining a light in to see properly, because the wires were not on the side of the display window. However they were bunched up a bit to the side. So I used small personal grooming scissors through the window to cut one of the coated wires.

Hurrah, no more beeping! All I have to do is keep it in my mouth for a minute or so. It displays the temp the next time it's turned on. So now I can keep my charting private.

Funnily enough, this stoner tutorial was the most helpful tutorial I read today, yet nothing to do with TTC :) https://www.marijuana.com/community/threads/step-by-step-guide-how-to-make-a-digital-thermometer-silent.261588/