Bluetooth proximity auto car door lock

Hi, I am after help/suggestions on. I am considering using a arduino to make a simple proximity sensor using Bluetooth to auto lock a car when out of range( small range like 1m or so), I do not want unlock just lock and I want to use a dongle of some sort so it just stays in my work bag. I have to park my car everyday and sometimes you think did I lock it.

Basically I see it work like this

Ignition on module becomes active with a timeout if it finds no bluetooth dongle after X amount of time. If it finds a dongle then it waits till dongle is out of range outputs to a relay to that will lock the doors. When the doors lock it will cut power to the module.

Can it be done cheap and easy?