Update: My device with this problem has been returned for warranty service for another problem, but I have been using Zoiper for the same VoIP service on a cheap Samsung Galaxy J1 with Android 5.1.1 and it has no problem changing between Wi-Fi and LTE.
It just makes a notification "(number) registered" and is completely painless and perfectly robust, unlike Zoiper on the phone with the problem (a Sony Z5 Compact that I paid 4 times as much for - not happy).
It looks like a device or OS problem at this point.