Sorry to hear of your difficulties.
I am in an equally barren network area in the New Forest. Mobile network signala are very weak and/or non-existent, hence the need for wi-fi calling.
Also our BT Broadband connection is very fragile and misbehaves with the weather - we still have copper wires on poles!
For the time being the wifi calling is working and I use it all the time, even with the landline doverted to mobile.
In spite of changing settings, re-booting etc I could never get a fix at the time, although it does seem to work at the moment.
I can only think that the combination of poor/unstable broadband and failures in the mobile network were a cause, but I could never prove it.
I suggest you raise a technical query with both BT and your network provider - get their wizards to investigate!
I am sorry you're WiFi Calling is disconnecting and hope your neighbour is OK and gotten the help they needed.
I can see you've tried some troubleshooting already, just to check do any other devices disconnect from the router at the same time?
Does WiFi Calling remain connected when using other WiFi networks?
Speak soon.
Leanne.