For others reading this thread - Just because all the pins are covered it does not mean they are all used
(see below though)
The best site (please think about supporting it via Patreon) to find out what is actually used is https://pinout.xyz/
and follow the link on the right ‘browse more HATs...’
Unfortunately in your case though, all you have left is ground and power display pins used
I would look to using something like a USB to Serial connector to connect from the Pi to something like the Arduino or ESP8266 series of chips and let that take instructions from the Pi and feed back switch status. You could even go WiFi and run MQTT on the Pi and ESP to do the communications for you if you do not want to be tethered.
USB to Serial adapters are very cheap and often sold as part of a kit or built onto the board.
You could look at starting with this board
and a suitable adapter. Give the good folk there a quick call or email and they will set you up - I’ve always had great and fast service from them
Need Pi spray - these things are breeding in my house...