Surely the reason the German police believe Bruckner killed her is because he filmed it.
No, I don't think that is the case. The German police read the chat logs, where the suspect wrote how he would like to 'get himself something small to use for days'. And whe challenged by the guy in the chat about this being risky he said something like: 'Not risky, if you destroy the evidence.'
So, if the police think this chat is evidence the suspect is responsible for child abduction, they will also conclude that his 'modus operandi' is to kill the child ('destroy the evidence') afterwards.
Also, not sure how she could still be alive in such a scenario, unless the guy sold her to someone else. For this, which would have been risky for the subject, police proably just don't have any evidence at all. Something like that may simply not be in line with what they know about the suspect. So, if they believe suspect did it, they consequently have to believe suspect murdered his victim.