So it turns out that no extensions are needed for the Sleep Button to be detected on my X200 ThinkPad (neither acpid.tcz nor acpitool.tcz nor thinkpad-acpi-KERNEL.tcz).
udev detects the button at boot and creates /dev/input/eventX (in my case, /dev/input/event0). After that,
Xorg automatically detects the button as an xinput device (both "cat /var/log/Xorg.0.log" and "xinput --list" confirm this). It is curious that a single button is treated as its own separate input device, but the udev/Xorg creators must have had their reasons.
Now all I have to do is to bind XF86Sleep to some action. I use sxhkd.tcz to manage my keyboard shortcuts, so will associate XF86Sleep with
sudo sh -c "echo mem >/sys/power/state" in ~/.config/sxhkd/sxhkdrc and all will be well
And the Sleep Button (Fn+F4) lived happily ever after.