I think I reproduced this fault...
So, something in the HC-06 while in un-paired mode messes up the power bus.
So, my only other option is to somehow make that default circuit for the Biss0001 less sensitive.
I dug out my BISS00001
Chinese night-light module and an HC-05. With the BISS00001 running on isolated 4.5V battery power and its PIR sensor covered, I got false-triggering when I dangled it in the middle of the breadboard containing the HC-05!
It wasn't exactly reproducible at will, but the effect was definitely there. Moving the module away from the breadboard stopped the false triggering completely.
There was no electrical connection between the two circuits.
So ... I conclude that the answer is not smoothing (or filtering), but
shielding. (And you owe that pretty little Vigortronix PSU an apology
)
Looking at the
photo of your board, perhaps you could add something like a piece of copper-clad PCB board at 90° to the HC-06 ... and connect it to GND. Something underneath the PCB to shield the relay might be required as well. This is definitely not my area of expertise, so hopefully you'll get some better suggestions.
(There are worse things for a BISS00001 to have to contend with than an HC-06 though. When I stuck my test module in the middle of my 3xESP8266 breadboard, it triggered immediately and stayed triggered until I removed it!)
One thing I did notice about
your circuit, was the arrangement of the caps. around the HC-06. As it stands, with them connected to the collector of the transistor, there is a little pulse at the output of the BISS00001 every time the transistor conducts. It only lasts about 380uS and is tiny (120mV). I doubt it affects anything, but connecting the caps. directly to GND gets rid of it.
As unrelated observation, I was impressed by how sensitive these PIR sensors are - to block it, I used a piece of paper folded multiple times, held in place by 2 layers of insulating tape. If it put my finger directly on top of the sensor, it still detects it!