Hi - I'm sorry to say I have no experience with streaming, or with using Steam. That being said, I did some searching and found lots of posts about getting the steam overlay to work, and very few about getting the origin overlay to work through steam. Most of the answers that are marked as solutions dealt with settings that are no longer found in Origin (like automatically close Origin when exiting the game). With all that being said, I did find this post - under the spoiler tag is a long solution for getting the game to work in steam, which it seems you have already accomplished. However, the bottom part of the solution suggests a way to get the Origin overlay to work.
They suggest to set the game options to run in windowed mode, making sure your resolution settings match your monitor settings. Then, exit the game, right-click the SimCity icon in Origin; under Advanced Launch Options > Command Line Arguments box, enter "-noborder -windowed" (without the quotes). I will assume you have to also check the box to "Enable Origin In-Game" under the General tab, since you want the overlay.
This post, which is from 2017, suggests that you can run one overlay or the other, but not both.
With all that being said, I realize that it may not be super helpful to have someone list stuff from searches that they haven't tried. I am paging and - can either of you help to get the overlay working with Steam? Is there a better forum to post the question? Any help would be appreciated.