If you want a motion detector, you could use a standard intruder alarm one (typically powered by 12 v) Measure the output when it is 'activated' and connect the device to the appropriate inputs of a UNI or Shelly 1 - There are several examples on the web of people who have done this (some with mains PiR - hence the Shelly1)
That said, I can't think of any reason why replacing the battery with a permanent supply wouldn't work in theory (try it at your own risk of course)