Steam overlay messes up black borders

[pas-de-2]

In Borderless and Fullscreen Upscaled modes, the Steam overlay 'overwrites' the black borders and leaves a smear behind whenever it shows up, like so:

Also happens in Warcraft II, and I assume other cnc-ddraw games.

Steam, for whatever reason, doesn't let you disable the overlay on a per-game basis for non-Steam games. A workaround is to point the Steam shortcut at a .bat file that launches the exe instead.