Hello,
For this project I would recommend BLE, since BLE has distance measuring right into it (build in the devices and is very reliable).
For a BLE module RN4020 would probably do just fine, it's low cost, but still with good range and reliable.
Since you want an Arduino prototype, I would recommend tests with Atmega328p (standard UNO) and when finished the development transfer to ATTINY45 or Atmega8A, both are compatible with arduino but are low cost ~0.7 USD in quantity for ATTINY45 and ~1.5 USD for Atmega8A, much can be saved in future mass production this way.
So I can offer you a PCB with Arduino compatible controller (your choice), with BLE module, with LED drivers, and power management all in one board as the final device should look like.
If you choose to prototype with arduino standard boards you would need to make a shield and when it runs, you start making this board, that I can offer you now.
If you accept my offer for direct design of the main board, for the same time as for a shield I will design the main board in size close or less than the shield and by that saving you money from the development process.
So the bottom line:
I can make your whole device with CPU, BLE, LEDs and power management in the time and the budget you are giving for just a piece of the prototype.
If you have interest or questions, please contact me.