There are 2 ways to deal with this. The first option only works if a command is used to return the piece to the private window (i.e., the player doesn’t just drag-and-drop it back on the private window). For this method, instead of directly triggering the Send to Location (or whatever trait you’re using to return the piece), you want to use a Trigger Action, and have it trigger the Send to Location, followed by triggering the Key Command for your Mask trait.
The 2nd option will work even if the player drag-and-drops the piece. This requires you to edit the properties of the Private Window instead. You can access the Properties window by double-clicking on the main entry for the map in the editor. You will need to change the last entry, Key Command to apply to all units ending movement on this map, to send the Key Command that triggers the Mask trait. The problem with this method is it will trigger when any piece is moved onto (or within) the map, so it could cause unintentional flips (of this piece or other pieces that use the same Mask trigger). You would probably need a Restrict Commands trait to prevent this trigger when you don’t want it (which could be complicated).