Mass Effect 3 has the same Origin issue, where if you launch the game .exe directly, Origin kills it, then relaunches it. That's why steam "loses" the game shortly after launch.
Luckily, Kotsu figured this out for us in this question.
Basically, you must add Origin itself as the non-steam game. This is because Origin requires that it be the parent process of Dragon Age, and if it isn't, Origin will kill the game and relaunch it. That makes Steam think the game is done, and you lose the overlay functionality. By linking directly to Origin instead, you keep the overlay on any game launched by that instance of Origin. Also, you can set Origin to exit after you close DA:I so that Steam sees you have stopped playing if you'd like.
Steps from linked Answer:
If you were hoping to access your Steam Overlay within the game, this will require a bit of a workaround.
Instead of having a direct shortcut to DA:I, you will have to create a shortcut to Origin:
- Clicking Add A Game in the bottom left of the library tab
- Add a Non-Steam Game
- Local Disk --> Program Files (x86) --> Origin --> Origin.exe
Make sure Origin isn't currently running, then launch it from Steam. Now, when you launch any game from this instance of Origin, you will be able to access the Steam overlay.
As you start the game at last with Origin, you may add your command lines there.
I can confirm the linked solution worked, and allowed access to the steam overlay. Also, since you need to launch the game from the Origin menu, you can use Origin's ability to set command line parameters (right click game, go to Game Properties...) to keep your MaxSimFPS settings.