How can I set QGIS to not mess up my TIFF images when I geolocate them?

I am using QGIS 3.10.0-A

I have some TIFF images of simple line maps. I worked on the TIFFs to reduce file size while retaining quality and got them to where I want (for example, I used Irfanview’s Batch Processing with 16 bit/4 colors (black and grays) and LZW compression, 300 dpi, 2947 x 4385 pixels, to get good looking lines with 408 KB file size). I have tried various ways of geolocating these TIFF files in QGIS and then trying to save the geolocated Tiffs, but every way I try screws with the image and makes it look bad, apparently lowering the number of colors to two (jaggy/pixilated lines like a FAX), or maybe using a different compression scheme (makes the lines look like fractals on the edges). And the file size goes up dramatically (much lower image quality, much larger file size).

I have tried using the Freehand Raster Georeferencer plugin to do the geolocating, and everything looks great until I use the plugin’s "Export raster with world file," which edits my TIFF and makes it look bad (and when I try to open the file in QGIS it opens at 0 lat, 0 long). If I try to use print composer to save the Freehand Raster Georeferencer layer as a GeoTIFF that also does not work.

I have tried using QGIS’s built in Georeferencer to geolocate the TIFFs, which locates them properly, but makes the lines look like fractals on their edges. I have tried to use print composer to save those rasters as GeoTIFF files, but it also makes the Tiffs look bad.

Trying to find a workaround I have tried to use the World file from the same TIFF, generated by Freehand Raster Georeferencer or print composer, with my good quality Tiff (by making the file names match except for the extension), but the TIFF does not geolocate properly. I read that if a TIFF has geolocation in its header then the World file is not used, so resaved the TIFF in Paint.net to be sure that was wiped clean if present, but still no joy.

It seems this should not be so difficult, which makes me wonder if there is a setting somewhere for how QGIS handles the TIFF image when it is georeferenced. To me, the default should be to not mess with the image, as people likely have it like they want it when they go to georeference it.

So, is there a way to set QGIS to simply add the Geolocation information to an existing Tiff file without changing the image itself?

I expect there is a way to do this with GDAL, but I am no wizard and would need clear step by step instructions.