How do I add Unix executables to the Parental Controls 'Allowed Apps' whitelist? One example is some of Karabiner's executables, I get popups saying "You don't have permission to use the application "karabiner_grabber." as well as karabiner_console_user_server and a few others. They all appear to be located in /Library/Application Support/org.pqrs/Karabiner-Elements/bin.

If I select 'Always Allow' on the popup, it just comes up again within about 15 seconds.

The executables don't appear under the list in Preferences > Parental Controls > Apps > Limit Applications on this Mac > Allowed Apps

I can drag them in from Finder, but they disappear if I exit the preferences then go back in.
You could take the approach suggested here.
Open the terminal, then switch to your child's user account with:
su - [username here].
Go to the bin directory you mentioned above:
cd /Library/Application Support/org.pqrs/Karabiner-Elements/bin
Then, give all users permission to execute all the files in the bin directory:
chmod a+x *