Have you tried applying the overlay after tying the brush to an enitity?
I don't think it's possible to place an overlay on some of the "dynamic" brush entities, like func_door.
Try using a decal instead of an overlay.
or maybe make the brush you want to apply it to the info_overlay's parent? it IS an entity, so you can set its parent. Dunno if it works, but I think it's wort trying.
As far as I know, there is no solution.
But if there's one, I'd like to hear about it since I'm using hacks for 3 years now.