In this post, I modified the ThingerWebConfig.h code and managed to connect on the Private Server, but from what I remember, I think it didn’t work for the Public Server. I would have to do more testing.
With thing.set_state_listener (link) it is possible to create a logic to change the SSID and password of the router/modem if necessary.
I used a push button to trigger WiFiManager’s restconfig function.