Accelerometer on doors and windows
The first thing to do is to mount the accelerometer as far out on the swinging end (not the hinge end) of the door as possible.  Then, of course, the accelerometer axis must be aligned perpendicular to the door.

High pass filtering may help, but that requires leaving power on.

Ultimately, you may not be able to do get what you want within the constraints you have imposed on yourself.  The obvious answer is to use a device appropriate for this application.  Not all things are possible just because you can write a spec for them.

"I bought a bushel of tomatoes instead of a lug wrench, and now I'm on the side of the road with a flat tire.  How do I use these tomatoes to get the tire off.  No, I insist on not walking to the hardware store around the corner to get a lug wrench."