As you can tell, the map stays. The problem is, its not perfect. Part of the pixel in the first image is outside the vertices. I used the modifier and an empty to control it, manual rotating, moving, and scaling the empty as needed. When I don't do that, the map looks like this: When it should look like this: How can I make the uv map stay like in the first 2 pictures without messing up the UV map?
Here is a setting with 3 empties as projectors on a cube.
Face: pointing to the cube's face. Z of the empty is opposite to the face
Right: pointing to the cube's right par. Z of the empty is going right (opposite again)
Left: same principle
Use "override image" option in the modifier.
Each empty has been moved and scaled to fit the image texture at the wanted location.
edit as unclear. If you don't use override option, just make so the empty are well oriented (minus Z pointing to the wanted face). It supposes that you have placed the UV map correctly before using the modifier.