You would need to setup the phone to run an app that calls outwards when the BLE triggers it either with a custom profile for BLE, or a declared profile on the BT SIG website: List of Bluetooth profiles - Wikipedia
The process of connecting the BLE will be up to your architecture choice of how you want the connection to be: constants, fast, low-power, etc.
Both of the methods you describe are feasible and implementable.
There are many devices that implement this, but most of the work, unfortunately, will be on the phone-app side of the devices.
The BLE device will merely send a button-press trigger signal to the phone, and the phone app/code will do the rest from the architecture standpoint.