I've got some similar to that. Those can be tricky as they get older they get brittle and do not flex like they once did. Mine have to go on the way they came off to fit snug.

Make sure your have tried it both ways. If you find it snaps in place use a marker to dot the cover and the fixture so you know which way to put it on next time.

Personally I don't think I would run it without the cover. Not that I think much will happen. I just don't think it's worth the chance. I would simply turn it off and buy a new fixture. Maybe it's time to add a led strip. They're cheap on Amazon.