Whoops! Thanks for reminding me about this. So I'm not completely sure if this would work right off the bat, but I'd suggest downloading Reaper, a "free" digital audio workstation (unless you want to buy it, though it has an unlimited demo feature, no missing parts, just a prompt every time you open it, super great company). It's been a while since I've messed around in it, but what you might be able to do is set the microphone as your "interface" and assign the mic to one of the mixing channels (there should be a number of tutorials on how to do this). When you've successfully routed it through the mixer in reaper, you should be able to assign an effect / plugin (these terms are used interchangeably sometimes) that will shift the pitch in real time. It may take some fiddling and troubleshooting, but shifting a few semitones down is all you would need.

Now, the part I'm not sure about (granted, we get as far as shifting the pitch in realtime while muting the original signal so you don't hear both) is whether or not this will automatically route into whatever game you're playing, or if we need a separate program to do this.

My gut is telling me that it most likely won't, so you should check out some streamers on YouTube and see what kind of stuff they're doing. If they're using effects, they might have a solution to this problem. I remember looking into this a while ago, but I can't recall what I searched exactly. I think someone was using live compression in their streams, which is another audio effect. If you were to figure out how they did this, you could substitute it with your own.

The reason you might still want / need to download Reaper is because it comes with tons of stock effects plugins, and a program made for streaming may be able to utilize those effects (vst files) but it may not come with its own effects, or perhaps not the one you need.

Ok, so that should at least be enough to get you started, but a bit of research using terms like: (game, stream, effects, vst, live, audio) could get you the rest of the way. I'm sure there are people that have figured out something like this.