I vote for covering the window.
Several of my older cameras have covers built in. I figure if people thought to include covers, back then, they must have had this problem, too. If it was a problem for them, it can still be a problem for us, too. Right?
I'd use a piece of gaff tape (or similar) and a small piece of cardboard to make a semi-permanent flap with a pull tab on it that you can open and close when you need it. Maybe not the most elegant looking thing but effective and it won't permanently alter or damage the camera. Remove the tape and clean off any left over stickum.
(The reason I often suggest gaffer's tape is because it has less tendency to leave stickum behind if one is careful.)