Don't know whether it helps but this is the wiring for same on my Roadster Andy.
![[Linked Image]](https://i.imgur.com/mHCszWA.jpg)
As you can see a part of the BCU is fed by a fused ignition switch (run position) feed to power the screen itself, with the momentary switch commanding the BCU to turn on heating for pre-set time or until it receives another momentary command to toggle off.
You will also note the switch lighting (dim or bright to indicate off or on) is directly under the command of the BCU, not the switch itself. I can see from the Plus 4 chassis loom diagram that yours is also BCU controlled, and your comments suggest it works in the same manner. It seems therefore the BCU or its connections are responsible for the problem not the switch.
Would be interesting to hear what Oakmere have to say on the issue. I would be tempted to say pulling fuse 10 in the body fusebox will kill all power to the screen and save your battery but for the fact it appears to be powered with the ignition off

ETA: have just checked and the Plus 4 connections to the BCU for control of heated screen are identical. If not the BCU then an intermittent short to earth on the orange trigger cable to BCU pin B14 would simulate a button press and cause the issue. Following on from this the issue could after all be an intermittent short in the switch itself or the PCB it sits on and the BCU is just blindly doing what it's told. Any signs of water damage? If not I think you have a strong claim with Oakmere.
Unfortunately none of this explains why the screen is ever energised with ignition off unless of course the BCU is at fault