Sorry to say, but it looks like support for this just isn't built in to git at the moment... Comparing this:
You'll see that the latter has various references to color (starting with including
color.h), and the former does not. That could be a starting point for you or someone to modify the git source (and submit a pull request or similar) to have this functionality.
I'm not sure why you're not having success with stderred, though;
git push does, I've just confirmed, send its output to stream 2 (stderr), so it seems like it ought to work, provided that it's working for other utilities.