I took a quick look at how to do this, and it seems like you can cross compile to Windows under Linux. Everything was going smoothly (although I've never used Go before) by just following these instructions: https://github.com/golang/go/wiki/WindowsCrossCompiling. In the end I ran into a problem with pixel. Seems like even that is doable though https://github.com/faiface/pixel/issues/53, but sadly I don't currently have enough hard drive space on my Linux system (docker pulls a 700 MB+ of stuff) to try it out.
Edit: Switched to a Windows computer and got 'er done: https://www.dropbox.com/s/pozgtof6v6s2942/Ice-w64.zip?dl=0. It was quite painless despite using Windows (although gcc or some other part of the build process had trouble with my username containing non-ASCII characters (Käyttäjä = User)) and being a first time Go "user". Go get & go build are lovely tools!