JOB: Simple timer project to kick things off

What hardware and communications to pick depends also on the means / skills that are or have to be made available for the building, operating and maintaining and the physical constraints for 'the button'. About the latter you have not elaborated much yet.

As @Ollie points out, Bluetooth BLE may not work across the whole space because of the obstructions. Therefore, at one point you need Wifi.

The lesser variation in hardware and communication you have, the 'easier' and more cost effective is it to build, operate and maintain your system.

If you do not have real physical constraints for 'the button', I would use only as hardware just Espruino-Wifi run by two standard/exchangeable 18650 LiIon batteries. All this fits in a small box you still can carry in you pocket. Espruino-Wifi is 30mm x 23mm (1.2 x 0.9 inch), 18650 LiIon is 18mm diameter x 66mm long (in inches: 0.7 diameter x 2.6 long). All in all about 1.5 x 3.6 x 0.8 inch. All buttons look the same, have the same software installed, etc. On power up, a sequence of (electrical) button presses makes the buttons to be what they supposed to be: button in rooms, button on bearers, 'button' in hallway as monitoring central station.

For display, you have multiple options:

Frugal option:

• For button:

  • for application/user feedback in the buttons, use built-in green and red LEDs
    
  • for communication/connection status, use built-in blue LED
    
    

• For central station:

  • for application/user feedback for monitor and control operator: simplest variation is making central station run as a Web server (and optionally also access point), serving the buttons as well as a Wifi device with display, such as a phone or tablet, for the central station display.
    
  • for communication/connection status, use built-in blue LED
    
    

If you want to have a dedicated display, you may connect (plug-in) a TDF or OLED or LCD display.

If you want to spend a display on all buttons, put an OLED or LCD display in every device. If you pick a TFT, it uses a bit more power, but has the advantage of having a (resistive) touch screen... and the things are cheap.

With an OLED or LCD -but not touch - in each button, a 4 way with enter button fits and you may make it even a 'game boy'...

Since batteries can be exchanged, you can have enough charged spares available, and to operate reliably, you just built 10..12 buttons.

As a last comment: take a look at upcoming Espruino Pixl.js - just too bad that runs BLE only - but would have all what you need... Adding the batteries and a cheap ESP8266 ESP-01 serially connected with the Pixl.js will also give what you need... more than you need... and it is really a cool thing... see the fun NodeConf badge by @Gordon ... you will enjoy and be inspired... (and choose how to move forward).