I think you're approaching this from the wrong direction. You don't HAVE to stream at the same resolution as the game you are streaming. It would be best to ask yourself what resolution you can stream at, and then stream everything at that resolution, even if that involved scaling up or down in one way or another. Games will have to be scaled up or down anyway when people fullscreen your stream, right?
Most streaming services have 16:9 players, and I'd venture a guess that most people have 16:9 monitors, so the aspect ratio you stream might look good for you, but might have black bars for everyone else.
I don't really see the problem with just picking your best resolution to stream at, and then just streaming things at that resolution. Aspect ratios might be inconsistent, but it's not like streaming at the aspect ratio of the game you are streaming is going to eliminate black bars.