After some unsuccessful googling, followed by loads and loads of digging and grepping through binary files, I stumbled upon a key in a .plist which, when written, appeared to make the system temporarily forget it had any updates to bother me with. Running:

defaults write com.apple.systempreferences AttentionPrefBundleIDs 0

fixed my issue for now on 10.14.1 (Apparently, you may also have to restart the dock with killall Dock, but I don't remember having to do so).

It seems to work up through at least Monterey (12), too.

If you have automatic checking for updates turned on, this might not work for you. Not for long, anyway. In my case, on a fresh install, I went to the settings page to turn off checking for updates, at which time it checked for an update before I could close the page. After running that command again it went away once more, though. Basically, turn off automatic update checks, or it'll come right back and you'll have to run the command again.

Re-running the update checker should undo this. One might be able to find that .plist (I believe it was in /Users/[username]/Library/Preferences/com.apple.systempreferences.plist) and set the immutable bit on it/give it read only permissions to prevent it getting modified, but I have no idea what the fallout from that could be. You'd probably not be able to change any other per-user settings anymore.

Edit: Some have suggested setting it like:

defaults write com.apple.systempreferences AttentionPrefBundleIDs '{ "com.apple.preferences.softwareupdate" = 0; }'

instead. This seems to work, and is the default value on a fresh install, at least for some newer versions - but it is also more difficult to type and ultimately has the same impact, so I still just use the top version and have noticed no ill effects.