MacBook Pro: Built-in keyboard and mouse unresponsive

I have a MacBook Pro 13", early 2015, with OS X 10.11.4 installed.

I mostly use it at work, with its screen closed, an external screen (via Thunderport), and an Apple wireless keyboard and Evoluent Vertical Mouse 4 – both connected via Bluetooth.

Before taking it home, I open the built-in screen, unplug the power cord and Thunderbolt screen cord, and then close it. I leave the Bluetooth keyboard and mouse on as I leave the office.

This usually works, but yesterday at home my Mac kept looking for the Bluetooth peripherals, and wouldn't respond to the built-in keyboard and mouse (except for the power button when kept pressed 10 seconds for a forced shutdown). Rebooting changed nothing, and I tried rebooting with the Disk Utility (Cmd + R) or resetting the NVRAM (Cmd + Option + P + R) and the pressed keys did nothing. The SMC reset (Shift + Ctrl + Option + Power) didn't appear to do anything either.

When I connected an Apple USB C keyboard, the keys were logged, but it would still look for an absent mouse and go nowhere else than the "Looking for a Bluetooth mouse" screen.

Back at work today, everything works: built-in and Bluetooth peripherals.

Any suggestions?