Can you use the pixel weighting values when exporting a Google Earth Engine image asset to a GeoTIFF?

The “weightings” are known as the image's mask.

if the absolute value of a pixel in code editor is 1000 and the pixel weighting shows as 50%, then the value exported to the GeoTIFF in my drive would be 500, whereas currently it seems to remain 1000.

You can modify the pixel values in this way by exporting image.multiply(image.mask()).

You might also want to try combining this with unmask and updateMask to convert to an image whose mask is always 1 (100%); this image will display within Earth Engine more like what you're wanting to export, but it won't modify anything but the mask.

Demo script:

// resample() ensures that the edges of this image will be partially masked by
// interpolation between masked and unmasked pixels, so it demonstrates the
// effects of what we're going to do to it.
var image = ee.Image('LANDSAT/LC08/C01/T1_TOA/LC08_001004_20140524').resample();

// Convert the image to fully unmasked:
var unmasked = image
    // Convert fully masked pixels (only) to zero.
    .unmask(0, false)
    // Convert partially-masked pixels (only) to fully masked pixels with the same value.
    .updateMask(1);

// Multiply the unmasked image by the mask; this gives the same effect
// as mosaicing or visualizing the original masked image on top of
// a zero/black image.
var multiplied = unmasked.multiply(image.mask());

Map.setCenter(-21.85161, 78.02296, 16);
Map.addLayer(image, {}, "Original");
Map.addLayer(unmasked, {}, "Unmasked");
Map.addLayer(multiplied, {}, "Multiplied");
Map.addLayer(image.multiply(image.mask()), {}, "Multiplied w/o unmasking", false);

https://code.earthengine.google.com/287cbf2109e1b8e2ed06a9d5aa1cbb54