That's one thing that Photoshop offers that is great for doing these types of logos.
Once selecting the # of colors when saving for web, you can then adjust the amount of dither with a slider. Or you choose a dither preset. Helped me a ton over the years. From there, you select the swatch to input the exact values as they are in TM**.
Not sure if Gimp can do that or not.
** Make sure you update the values in TM for colors first, and use that value before saving the file in the image editing program you use. Otherwise, they won't match whatever you're using as a reference (NOSE, for instance).
This will prevent oddities when importing your image or logo.