Hey, the same happened to me, but with just an Arduino and a BLE module. I suspect the battery was old and I have replaced it since then, but you need to decide what kind of “risk” you want to take and so if you want to disconnect only the iPhone or the entire thing, including Arduino itself.
I couldn’t figure out how exactly you are powering the devices (regulator, step-down, etc) but having the Arduino turn the phone off if the voltage is low is relatively simple. Have the Arduino sense the battery voltage through a 1:2 voltage divider (divide by 3), then connect the phone’s power through a P-channel MOSFET controlled by the Arduino. Now the Arduino can turn the power supply on and off.
My plan is to turn the entire device off which will require a button press to turn it on again. I don’t have the design public yet, but send me your email address to http://fochica.com/contact-us/ and I will send you a draft. The design is based on https://makerself.wordpress.com/2014/12/23/power-circuit-redesigned/ which is a great resource you will find interesting.
Have you measured how much current your devices take? iPhone vs Arduino power consumption? Do you have a schematic for your circuit?