You can of course, but the ImageIO API is terrible. This may get you started:

/** If alpha is true, returns an RGBA buffer, else returns an RGB buffer. */
static public BufferedImage pixmapToImage (Pixmap pixmap, boolean alpha, @Null BufferedImage reuse) {
	DirectColorModel colorModel;
	if (alpha)
		colorModel = rgbaColorModel;
	else
		colorModel = rgbColorModel;

	if (reuse != null) {
		if (reuse.getWidth() == pixmap.getWidth() && reuse.getHeight() == pixmap.getHeight()
			&& reuse.getColorModel() == colorModel && reuse.getRaster().getDataBuffer() instanceof BufferRGBA rgba) {
			rgba.setPixmap(pixmap);
			return reuse;
		}
		reuse.flush();
	}

	DataBuffer buffer = alpha ? new BufferRGBA(pixmap) : new BufferRGB(pixmap);
	SampleModel sampleModel = colorModel.createCompatibleSampleModel(pixmap.getWidth(), pixmap.getHeight());
	var raster = new WritableRaster(sampleModel, buffer, new Point()) {};
	return new BufferedImage(colorModel, raster, false, null);
}

static private final DirectColorModel rgbaColorModel = new DirectColorModel(32, //
	0xff000000, 0x00ff0000, 0x0000ff00, 0x000000ff);
static private final DirectColorModel rgbColorModel = new DirectColorModel(24, //
	0x00ff0000, 0x0000ff00, 0x000000ff);

static public class BufferRGBA extends DataBuffer {
	protected ByteBuffer pixels;
	int width, height;

	public BufferRGBA (Pixmap pixmap) {
		super(DataBuffer.TYPE_INT, pixmap.getWidth() * pixmap.getHeight());
		setPixmap(pixmap);
	}

	public void setPixmap (Pixmap pixmap) {
		if (getSize() != pixmap.getWidth() * pixmap.getHeight()) throw error();
		width = pixmap.getWidth();
		height = pixmap.getHeight();
		pixels = pixmap.getPixels();
	}

	public void setElem (int bank, int index, int value) {
		pixels.putInt(index, value);
	}

	public int getElem (int bank, int index) {
		return pixels.getInt(index << 2);
	}

	public void toInts (int[] ints) {
		IntBuffer buffer = pixels.order(ByteOrder.BIG_ENDIAN).asIntBuffer();
		if (ints.length != buffer.remaining()) throw error();
		buffer.get(ints);
	}
}

static public class BufferRGB extends BufferRGBA {
	public BufferRGB (Pixmap pixmap) {
		super(pixmap);
	}

	public int getElem (int bank, int index) {
		return pixels.getInt(index << 2) >> 8; // rgba -> rgb
	}

	public void toInts (int[] ints) {
		int bytesPerLine = width * 4;
		var line = new byte[bytesPerLine];
		for (int i = 0, n = width * height; i < n;) {
			pixels.get(line, 0, bytesPerLine);
			for (int x = 0; x < bytesPerLine; x += 4) {
				int r = line[x] & 0xFF;
				int g = line[x + 1] & 0xFF;
				int b = line[x + 2] & 0xFF;
				ints[i++] = (r << 16) | (g << 8) | b;
			}
		}
		pixels.position(0);
	}
}

I think I got all the app classes. The rest are imported from ImageIO and libgdx.

The GIF format is simple, if very old. However, making a GIF can be complex because of temporal quantization: reducing to 256 colors over multiple frames, without the colors flickering across frames. I suggest using gifski if high quality is important. There is a Java wrapper that we created or you can run it from the CLI. Gifski makes high quality GIFs, but they are large. If you want smaller GIFs, you will have poor quality. With the Java wrapper you don't need ImageIO garbage.

Your error is:

Unable to read file for extraction: gifski-java64.dll

Gifski has a static initializer and SharedLibraryLoader loads the DLL:
https://github.com/badlogic/gifski-java/blob/master/src/main/java/com/badlogicgames/gifski/Gifski.java#L9
SharedLibraryLoader readFile reads the file from a JAR file named gifski-java or the classpath:
https://github.com/badlogic/gifski-java/blob/master/src/main/java/com/badlogicgames/gifski/SharedLibraryLoader.java#L117
Since the static initializer does new SharedLibraryLoader().load("gifski-java"); the JAR file it looks for by default is gifski-java. Maybe that should be gifski-java.jar? Anyway, you can have it look on the classpath if you call new SharedLibraryLoader().load(null); before using other Gifski classes. That probably fixes it for you.

Instead of PixmapIO.writePNG you can get the RGBA pixels from pixmap.getPixels() which returns a ByteBuffer. The pixmap is RGBA8888 so each 4 bytes stores RGBA for one pixel. Pixels are stored in rows that are each the width of the image * 4 bytes.

You can send the bytes without encoding them as PNG. That is probably best, but if you must send PNG then you could subclass FileHandleStream to implement write and pass it to PixmapIO.writePNG. That way the PNG is written to the stream you return from write.

Using both Qt and libgdx seems quite heavyweight and a pain to synchronize and pass data via IPC. What is a "hollowout transparent window"? The LWJGL3 backend for libgdx has a setTransparentFramebuffer method:
libgdx/libgdxblob/master/backends/gdx-backend-lwjgl3/src/com/badlogic/gdx/backends/lwjgl3/Lwjgl3ApplicationConfiguration.java#L171-L175
Some discussion here:
libgdx/libgdx6852

Transparent windows are pretty low level, as is making clicks go through to the windows below. You may need to use OS specific APIs. It should be possible with libgdx alone, but will take some doing.

I've never done it, so I can't be of much help. Here's ChatGPT4's advice:
https://chat.openai.com/share/2d936ffd-7af4-43aa-899f-770c80cf7d54

You probably would use LWJGL3 for the backend with libgdx. LWJGL3 uses GLFW, so you can look at the backend and at GLFW for how the window is created. You can get the window handle from LWJGL3 or GLFW to make changes to the window they created.

You could also drop down to creating your own window that uses OpenGL and render with spine-cpp, you'd just lose all the nice things GLFW, LWJGL3, and libgdx are doing for you.

What you are wanting to do is pretty tricky, so will require a lot of legwork and experimentation.