Use the other tool "Raster conversion - Translate". There you can add additional command line parameters which are documented in https://gdal.org/programs/gdal_translate.html.
In your case you probably -ot Byte.. You need a rather new GDAL for signed byte, see https://gdal.org/development/rfc/rfc87_signed_int8.html
The Save as tool does not work because it has an UI only for the creation options, but selecting pixel type does not belong to the creation options of the GeoTIFF driver https://gdal.org/drivers/raster/gtiff.html.
