Cut out part of an textured surface

I have an assembly of a flat landscape with various buildings on it.

I have inserted historical maps from various years. These maps are made from a flat panel where I have added the image of the map onto a flat surface by adding the image as a texture. The size of the flat panel is scaled to fit the map image. Now to my problem - I want to cut out the more interesting parts of the map but when I do this, the texture-image distorts to fit the remaining cut-out.

Is there a way to firmly "glue" the texture to the surface so that it does not distort when I make a cut-out?

The sample file included is from a local village map from 1701.

My present work-around is to leave a small part at the bottom left corner and at the top right corner.

Leaving these corners makes the texture remain unchanged but I think this solution is a bit primitive.

If you take the attached file and resume the suppressed feature you can see what happens.