I leave a working amount of paper in the safe with the rest in the freezer. My safe is big enough to leave a box in it so I use an old paper box and bag to hold the paper and the whole thing goes into the safe. Over kill but the main thing is it's either closed right or it's very obvious it's open. I also never open the safe with the lights on. If it's empty I leave it open so it's obvious it's empty.
I'm likely to build a paper drawer into the next table/storage shelf I build. From pricing larger paper safes I think I can build a table for less money then a good larger paper safe would cost me.