good news. I just bought Affinity Designer for WIndows, installed it, loaded one of my 8-bit greyscale images. (actually a 24-bit RGB where all values in each channel are the same so it looks grey). I then immediately re-exported, and went beyond the default presets. My original was a PNG and the presents did not show 16-bit but if you look for the extra options, you'll notice there is also a 16-bit greyscale option, ditto for Tiff and likely for others I haven't tested.
So at least in theory this should work. It even offers extra smoothing for that conversion, great!
Does your game engine take 16-bit grey in several formats? (tiff, png,...?). Should all just work perfectly.