Yep. You have a mobile i3. The i3 line is a very budget-conscious CPU, to put it nicely. Also, the Elgato compresses video before passing it over the USB 2.0 interface, so is going to use a chunk of CPU as well for decoding after it actually gets into the system, prior to OBS intake, compositing, and encoding for sending out to the livestream.
Try downscaling to 360p or 240p instead; only 1000kbps is probably going to end up not being able to support much higher anyway.