While tracking down IE quirks and trying to gain rendering performance in the overlay, I tried to use a transparent PNG instead of using an opacity filter, since it generally renders faster than opacity. And it does ! Everything really runs smoother and is less CPU intensive in all versions of IE.
Long story short : I ended up by simply using a transparent png for all browsers.
Incredibly, it renders faster in Safari and Firefox too ! Less obvious, but anyway.
To test this, open one window with the patch, one other without, and resize the window rapidly while the overlay is displayed. The module page is a good test. If anyone has a more scientific approach... :)
Bottom line : this brings big performance gains in overlay rendering.
Put the overlay-bg.png image in /misc.