This is an old question, so I'm guessing that tom37 may have moved on by now, but I think I have an answer for anyone else with the same problem.
For reference, here is a view of a surface grid using a perspective camera.

Now let's say that we want to render a portion of this current view, but to the entire screen. Let's render the top-left quarter of the image, outlined in white.

If we move the camera, then the perspective will change. We can use scissors to mask the rest of the image, but we want that portion to take up the entire screen. Let's try changing the viewport.
public class PerspectiveTest implements Screen {
Viewport viewport;
Camera camera;
Sprite background = new Sprite(new Texture("grid.png");
Stage ui;
SpriteBatch batch = new SpriteBatch(250);
public PerspectiveTest () {
camera = new PerspectiveCamera(60f, 16, 9f);
viewport = new FitViewport(8f, 4.5f, camera);
ui = new Stage(viewport, batch);
camera.rotate(60f, 1f, 0f, 0f);
camera.translate(12f, -5f, 4f);
camera.update();
background.setSize(32f,32f);
}
@Override
public void render (float delta) {
Gdx.gl.glClearColor(0,0,0,1);
Gdx.gl.glClear(GL20.GL_COLOR_BUFFER_BIT);
int x = 0;
int y = -Gdx.graphics.getHeight();
int w = 2*Gdx.graphics.getWidth();
int h = 2*Gdx.graphics.getWidth();
Gdx.gl.glViewport(x, y, w, h);
batch.setProjectionMatrix(camera.combined);
batch.begin();
background.draw(batch);
batch.end();
}
// There are a bunch of empty overrides here that I have left out.
}
The important line is Gdx.gl.glViewport(x, y, w, h);
The viewport usually starts at x = 0 and y = 0 and extends to the full width and height of the screen. If we want to see only a part of that original viewport, we need to change both the size and the starting position. Here, I have moved the viewport down by one screen height and stretched the viewport to be twice as large in both directions. Let's see what happens:

The camera hasn't moved, so the perspective is the same, but we're only looking at what was the top-left corner of the original view! This topic had been giving me trouble for a few hours. I read this solution to a similar problem, and I realized it could be applied here.
To focus on any different portion of the viewport, simply choose x, y, w, and h accordingly. If you're going to do any more rendering in the normal fashion, make sure to reset the viewport with the original x, y, w, and h values.
I believe that the other three quarters of the viewport are not visible but are still rendering. It would probably be wise to use scissors in conjunction with this method in order to save resources.