You can do that in Python Console of QGIS by using a QgsRasterPipe object (pipe) for setting a renderer clone of the image employed as active layer before to use the 'writeRaster' method of QgsRasterFileWriter class (you don't need gdal_translate).
I used the following code:
layer = iface.activeLayer()
extent = layer.extent()
width, height = layer.width(), layer.height()
renderer = layer.renderer()
crs = layer.crs().toWkt()
pipe = QgsRasterPipe()
file_writer = QgsRasterFileWriter('c:/pyqgis_scripts/output2.tif')
To test it, I loaded a raster dem (sample_dtm.tif) at the Map Canvas of QGIS and then, it was rendered as a singleband pseudocolor layer with 5 classes (see below image):
After executing the script, the rendered raster (output2.tif) was saved with the espected renderer; as it can be observed in the below image (compare the thumbnails of sample_dtm.tif and output2.tif) when I load output2.tif raster at the Map Canvas. The renderer was effectively cloned in output2.tif raster. The code works.