I've messed with origin settings before when we were looking into disabling auto-updates for users when they install texture mods, as origin was constantly breaking ME1 and ME2 users. Digging into this, the game ID on origin is DR:230773600_t. Inside of your appdata roaming dir, there is Origin folder. In this folder is local_<hash of some kind>.xml, and in here are settings. One of the settings is called OigDisabledGames, and when disabling OIG, the value listed above is added.
<Setting value="#//DR:230773600_t" key="OigDisabledGames" type="10"/>
When the game is launched, it's launched through a program called EASteamProxy.exe, which then calls MassEffect3.exe and injects the Steam Overlay. I'm wondering if something in this launch process is ignoring the settings, since normally it's launched by Origin rather than the proxy program. I poked around a bit with [Origin] IGO32.dll, but Origin won't even startup if that file is missing. (IGO is In Game Overlay).
Personally, I don't have this issue on Origin or Steam. But from my testing, the 'disable Origin in-game overlay' works in the UI, but it doesn't actually apply to the game that's launched, so it's an Origin bug and not really a game bug. Origin in game overlay however has always been poor. Ever since they changed to this webapp/qt interface, Origin's performance (including overlay) has been terrible. The old UI interface was pretty fast before they axed it. I would trade an uglier interface for a much faster one.
I doubt it would work, but in ME3Tweaks Mod Manager's ASI Manager there is an ASI called Origin Unlinker by a developer named ErikJS. It runs after Origin decrypts and runs the executable and unhooks origin from the game process - I wonder if this would work in this instance.