JOB: Simple timer project to kick things off

This is how I'd approach it, at a high level.

Since I'm unsure how the Pucks' signal would fare over the distances and through walls I'd suggest each of the four rooms has Puck button and a Raspberry Pi Zero W acting as a receiver. Pucks could be carried in a similar fashion to your remotes around and between rooms.

Each Pi Zero W could listen on BLE for the Puck associated with its room, or probably better, to all Pucks to ensure good coverage. Pi Zero W boards are all connected via LAN.

Puck button events are relayed to Pi Zero W. A received confirmation could be sent back over to the puck over the same service. - probably UART would suffice to simply light up the LED. The Pi Zero W relay events over MQTT (or similar), to a server application on one of the Pi Zeros (primary) attached to a display running a web or console based UI there.

Second display probably runs another Pi Zero W as a slave and listens for update signals from primary to update it's own web or console based UI. Websockets will do here.

There's a starter for ten. Maybe others can finesse the approach.