We operate fixed line, no mobile service. So we know the exact location
(postal address + building identifier (EGID)) at the time of the call.

It's also not about routing. Calls to the nearest PSAP are
source-routed. So if a customer dials 112 we translate this to the
nearest PSAP operating emergency service 112.

Are you developing ESRP (Emergency Service Routing Proxy)?
https://www.dec112.at/en/dec112-what-is-an-esrp/

If not, then you need to send the call to ESRP and provide the user's device location.
Who is provide ESRP in your region?


I'm just anticipating what could go wrong or cause delays in the
chain to find out the best way to react to such issues.

Kamailio => HTTP => HELD-Server(inhouse) => HTTP => LIS-Server

If you are a fixed-line carrier then you always know where the user's device is located. Just required to populate your LIS database.
All timing looks for me predictable.


PS: What I could not find in the Swiss NG112 documentation: If
transmitting the location to the LIS server fails. Is the Geolocation
header just ommited? Or is there a way to tell the PSAP there was a
failure transmitting the location to the LIS Server?

I am not an ESRP developer, but why do you need a dedicated HELD server? Why LIS is not enough?
https://www.rfc-editor.org/rfc/rfc5985#page-5
image.png

In this case, your LIS server should contain static information about the user's device location. And case "If transmitting the location to the LIS server fails" will never happen.