Okay, I found a possible temporary workaround for everyone having the same issue:
1. Add the texture normally via the GSX Customization Tool
2. Go to %APPDATA%\virtuali\GSX\texture\handling\ [YourNewHandler]; open "rules.cfg" and add the line "couatl.basecolor = [YourColor]"
[YourColor] being the basecolor you need (army_green, blue, cool_grey, cyan, full_blue, full_cool_grey, full_cyan, full_orange, full_red, full yellow, green, orange, red, yellow)
3. Install GSX again via the Setup from FSDT. Do not deinstall GSX, but just install the Setup over the GSX you already have.
The name of your new handler will now appear in the GSX Customization Tool. The handler has the correct basecolor you have chosen.
In addition, if you now add a new texture, the dialogue with the basecolor is back. But GSX still chooses a random color, until you install it again.