Getting the battery pack opened was not as easy as I thought. The visible tabs put up quite a fight, but I was able to open it without damaging anything. To my surprise the battery pack had no balancing circuit at all, despite it being 18650 Liion cells. Only Plus and Minus and a temperature sensor are on the connector of the battery.
Since I had some batteries from a Thinkpad Battery at hand, which suddenly died on me, I was planning to build a new battery out of those. I discharged and charged all cells, matched capacities aaaand completly forgot about internal resistance... Since the vacuum is pulling 200W form the 6S battery pack, I should have been concerned by this... As an improvement to the original battery pack I included an battery protection and balancing circuit I found on Ebay. I also bought a cheap pcb spot welder which I connected to a car battery. It works quite well!
I performed the first charging of the new battery pack on an old grill in front of the house. If it goes up in flames, at least it's outside, but everything went well.... expect.... it still showed the same error. So I tested out other known errors of the dyson DC16, but everything seemed to be working fine. Then I had the bright idea to actually measure the voltage of the battery pack under load and surprise surprise the voltage brakes down. So I now had two dead battery packs.