Perspective corner pin, but for stabilization

If you want to stop all camera movement and correct perspective issues on a tracked rectangle, you can combine corner pin tracking using Mocha AE and CC Power Pin. Take a look at these two tutorials. The first one uses the new version of Mocha AE; the second uses the original version of Mocha AE that has a different way of getting the corner pin data into After Effects. 

Try this workflow if you want to remove all motion from the video and not reshape it. This workflow will remove all the camera motion from a shot with some static geometry in the scene.

If these workflows solve your problem, I need a better explanation of your design goals.

You could potentially unwarp the whole thing by using the Reshape effect and two masks. Other than that it sounds like you may want to have a look at the advanced corner pin in revisionFX RE:Map plug-in suite which offers such functionality right out of the gate. 

Mylenium 

Option 1:  Try Warp Stabilizer to see if it does what you need it to.  It's by far the easiest way to stabilize footage in AE, but might not do quite the job you're after, if you want rock-steady stabilisation.

Option 2:  Use the legacy Stabilsation tool in the Tracking window in AE.  You can use the LCD screen as your tracking object if it's clearly defined within your footage.

This article gives a bit more detail to this discussion:

https://www.schoolofmotion.com/blog/stabilize-footage-in-after-effects