Hmm, you will have to re-assign the background image to your inventory/menu windows when you are about to display them.
Currently, to display the inventory you do something like Game.InventoryVisible = true; (in game_daemon.script). You'll have to replace it by something like:
var InvWin = Game.GetInventoryWindow();
InvWin.SetImage("path\some_background.sprite");
Game.InventoryVisible = true;
Similarly the menu window, except the first line, since you already have the menu window stored in a global variable (WinMenu by default).
WinMenu.SetImage("path\some_menu_background.sprite");
WinMenu.Visible = true;
I can't test it right now but it should work this way.