Ok, so here is an update on new battery registration with the Schwaben tool.
The tool definitely cannot store the mileage for the battery change, it says registration is successful, but it always shows the same mileage numbers when reading out the details afterwards. So to that extent this does not work.
However, it does seem to be able to reset the battery charging algorithm. I say that because after using the tool my start/stop function worked again, which hasn't been working for at least two years, as the old battery slowly got weaker. (And yes, I normally switch start/stop off, like everyone else.)
Bottom line: the tool gets the job done in the sense that it resets the charging algorithm. It cannot store the actual mileage numbers etc, which is a shame but really secondary.
I'll keep the tool now.
