Why does a motion detector provide enough power for an LED but not a class 2 transformer?
The problem is the electronics used by the motion detector modifies the since wave. Most power supplies today are switching supplies most will not function with a modified since wave. An old school transformer that steps the voltage down and a full wave bridge to change it to DC would do the job. I have a battery powered motion activated sprinkler close to my pool to scare ducks and geese away that has lasted for years with fresh batteries every few months. Like this. Not an endorsement but the one I have is several years old and still works.