For anyone who's followed this thread so far -
To determine whether the problem was with my hardware, I ran the lxde iso of Debian 9 live. In that environment, all different forms of beep work without any problems on my laptop. So it appears that the version of Debian 10 I had installed (debian-live-10.3.0-amd64-standard+nonfree.iso) is not configured to support beeping on my laptop.
Since I depend on that beep, I'll be reinstalling a different version of Debian 10. If it all works, I'll be marking this thread as solved.