Actually, upgrading to an i7 will help immensely in streaming. In that case you wouldn't need a new motherboard as you simply get an used CPU that would fit in your motherboard.

I've done that with a meager 2600K, coming from a 2500K, overclocked that to 4.5 GHz and it still runs a LOT better than my 2500K to the point I can use a slower CPU preset now for some games.

In my case it's actually not THAT much of an upgrade, simply because upgrading my entire system will give better results, but this is simply a temporary upgrade until I get a completely new system.

But in your case... you still have a 4th generation Intel CPU. Meaning it's way more powerful than my second generation. So if you're lucky with getting a cheap i7 of that same generation, if able, to overclock, and you're still good to go!

Hell, not being able to overclock is not that bad. But it would result in better performance.

Also lowering singleplayer games FPS to 30 helps lessen the load of the CPU. In multiplayer games it's basically not a good idea, since you need the responsiveness; but in singleplayer games, it's merely something you'll need getting used to.

Many modern games already use more than 4 threads, so you'll not only get better stream performance, but also better game performance (raising the minimum frames, creating a more fluent gameplay with less stutter)!

In the end, a capture card wouldn't work with a single PC. A capture card grabs the image from the game, and your own CPU still needs to encode that image back to Twitch. But instead of doing it directly, you're now using a capture card.

It won't help in the end, except when you would have a console or a second gaming PC where you would want to capture the game from.

So a CPU upgrade is the only option in this case (or NVENC, but at 2000 bitrate that's not really recommended).