Games : How do I save on the PS4 using another account?

I recently bought a dancing game for the PS4 so it could be played for parties and visitors, but I don't want it to appear on my Trophies list, not to mention that the PS4 takes photos now once you get a trophy.

I set up another account solely for just that game, but when I tried to turn off the console, it's displaying a Save game message, saying that I should back up my saves somewhere.

Shouldn't it be saving on the hard drive? How do I make another account save a game's progress on the console?