Depending on what sort of finish you require, Gloss, Satin, Matt depends on what you finish it with, I presume your Poly is gloss, if it is then you will not be able to denib after your final coat if you want a gloss finish, you will have to apply in a dust free environment, room left with a closed door overnight and your box brought in, sprayed and left alone without raising any dust.
If you want a satin finish then the best way is to apply some wax over whatever base you apply, sanding sealer, oil varnish etc, you can then denib as you apply the wax with something like a Webrax pad, grey colour 1500 grit (although it is actually finer than 1500) is the cheapest way, it can be used lots of times and doesn't clog, cut one pad into inch squares and store the unused bits in a bottle / tub
Wax gives a nice tactile feel to your box, Woodwax 22 or a Microcrystaline wax is good.
I would try two or three coats of Danish and wax, if you get a run with your spray poly it's going to be a jumping up and down and teeth gnashing session.
You could finish it with Danish only but you will have to apply your first coat thick and let it soak in, subsequent coats are applied with a dust free cloth very thinly, you may need four, five or six to get a good deep sheen depending on the timber, I would still apply wax over that though.
Word of advice, don't use water on your box if you are using oil based finishes, it may get under your finish and make grey marks and cause other problems.
Andy