Apologies for the late reply @WallyR @Pierre84000. Thank you both for your answers. I looked into both methods, and I will comment on each method below.
@WallyR It took a while to test your suggestion, because network-tinkering with even the shortest offline time is typically met by strong opposition from several family members, so I needed to find an appropriate time slot.
Your method lead to a success. I was able to add the Eve Energy to the thread network and matter using HA and the S200. It was actually even a smooth experience, and also the current/voltage/power/energy readings worked out-of-the-box.
However, I managed to do this only by completely disabling the native IPv6 which I get from my ISP. While this worked, it is not a permanent solution for me, as this impacts the operation of other services on my home server too much. I got all this to work on HA and the matter-server running in docker containers on my Arch Linux server. I am happy that this is working, as I want everything to run on my one server, instead of having multiple RP4s floating around in the house for all kinds of services. That is why I did not even try this on the test installation on the RP4 I mentioned earlier.
In any case, before I got started with HA/matter/thread, I read about the network prerequisites, in particular with respect to IPv6, and tried to meet all of them. In fact, I think everything IPv6 is set up correctly, but the “solution” was to disable IPv6, so, I was somewhat disappointed.
@WallyR this is by no means a complaint to you, on the contrary, I am happy you lead me to a “solution”, so thank you again. This is rather a confirmation of what you said in your last message about IPv6+HA.
If there is any way to get this to work while IPv6 is enabled, I would greatly appreciate any input and I can help by providing logs, etc.
@Pierre84000 Thank you too for you message. I tested this, too, but things failed very early. I actually tested this first (because it does not require network interruptions), and I tried this one on the HASS OS test server on the RP4 (all up-to-date).
As it turns out, I cannot connect to HA OS on ws://IP:5580/ws. I suspect that the current version of the HA OS on the RP4 does not allow access on that port outside of localhost? I guess I can allow this via iptables, but as the other solution worked, I did not investigate any further. Also, I only used the RP4 (with bluetooth) for testing, my goal is to use HA on my home server (without bluetooth).