So, forgetting about setting the FG color, the way to create a surface with a precision smaller than 1/256 in any component in GIMP 2.10 is using the painting or color tools.

You can do this, and use the color-picker tool to check for success:

1. use at least 16bit precision
2. fill an image with black;
3. Set your foreground color to the lowest possible graylevel (010101 hex color on the color dialog)
4. use, for example, the bucket-fill tool and set the opacity to desired level (25% for 1/4 of 1/256 gray, (1/1024 gray)).
5. Use the color picker tool to check if you got to the desired level (1/1024 gray will show 0.000076 for each component - while the 8 bit gray #010101 is 0.000304 for each component).
6. Convert the image to "16 bit integer precision" back before exporting to .PNG file - I further tested and found out that attempts to save a 32bit FP precision image as a PNG file will result in an 8bit png file (all work above is lost). But if the image on GIMP is 16bit integer, then the PNG export generates a correct 16bit PNG. (Saving as native .xcf file format will preserve the precision, of course)