Not really sure why changing the base resolution is a bad idea.
You can simply set 1920x1080 as your custom resolution to get a 16:9 output stream! BUT, what I would not recommend is to then stretch your source to fit this aspect ratio. Instead use the black space and add stuff to it.
Er... 1920x1080 is smaller than 1920x1200.
What I'd recommend is going up to the next size up 16:9 ratio. Set your base resolution to 2560x1440, stretch your sources up to fill the area (but keep the aspect ratio). You'll then have black bars on the sides. You can slide your game video over to the side and populate the now-unified columnboxes with your webcam, chat, artwork, anything without being subject to the low-quality downscale that comes with sizing a source
down in the viewpane. You can then use the resolution downscale dropdown to easily hit 720p. It won't get you a 1080p stream, but it'll easily get you a 720p stream at a 2 downscale (recommended max for non-partnered streamers on Twitch), or 960p which most people won't be able to tell the difference on. Problem of course being that you will never be able to stream at native resolution, which gives the best results.
But yeah, that's one of the big down sides to running with a 16:10 monitor. Matching aspect ratios can be difficult to deal with cleanly.