I make bluetooth devices. I use the CC2541 soc from Texas Instruments. I've made a few similar projects
I think what you want is an app that listens for a particular device and once the button is pressed on the device and on the phone it will show the time, date and mac address. I've made an app similar to that, but they were used for fitness.
So, what I would do for your project is setup a corebluetooth compiled app on an iphone. I would then set the corebluetooth to scan for a specific device type. The scan is called scanforperipheralswithservices. You can look it up if you like. The scanforperipheralswithservices can look for a specific service or device. I would probably need to have the device in my hands to find out what services it supports. If you wanted me to design the bluetooth button then I could just use that service. Anyway, after the app finds the device it would connect to it and then wait for the device to transmit the button press. The button press is called a characteristic. If the button press characteristic and the app's button are pressed at the same time within some reasonable value then the date, time, and mac address would be saved. I would probably save them permanently in the NSdictionary. The NSdictionary keeps data even if the app is closed.
Feel free to contact me with any questions you may have. Seems like you have a good idea with this device. I have some bluetooth devices here you can have if needed. They have buttons.