Auth without internet connection

Yes, your are right - just WLAN.
Looking at the documentation I found that I have to connect to the “discovery service” - which is at the Nuki Site.
And there I can enable / disable /auth on the bridge.

And that’s the problem.
Or is it possible to (one time) enable it having the bridge connected to the internet and later only use WLAN. With new “paired” devices I just have to press the button on the bridge - nothing more.

Or can I totally set the bridge up with the app, without any need for the “discovery service”?

By the way - I checked your instructions here: Bridge discovery & HTTP-API activation

This is also a problem at one of my sites. “Devices” are routed to the internet via a separate network.
So the bridge would reach the internet via (just an example) 189.xxx.xxx.110 - and “Computers” will reach out via 189.xxx.xxx.120.
Internally they can reach each other (on some ports) via local IPs like 192.168… / 10.xxx.xxx…

This works here for an example with LIFX bulbs without any problem.

But reading your post I guess this would fail.

And I also think problems may happen if anyone else with my provider (dynamic IPs) also uses the bridge.
Not very likely - at least as long as not a lot of households use the bridge - but in theory any other client from my provider can get “my” IP address just a few hours after I got assigned a new one.