I would generally avoid the softer "home store" oil based poly's as they're more likely to get goopy (some react to oils at least some and some candy has some oil so.. might eventually be a problem).
I think one of the harder poly/cured finishes would be fine.
Waterlox or Behlens would be ok.. ish.. but you'd need to air the darn thing out for a month before I'd feel comfortable using it. I think you'd have similar issues with the various wiping varnish oil/varnish compounds in that small drawer.
I think my preference would be one of the better water based poly's like General Finishes Enduro-Var or something similar. It cures super hard, quite quickly, and doesn't really have any residual odor. It's very lightly ambering. You can put it over an oil stain base IF you let the stain base fully dry, I'd probably prefer a water based stain if you needed to make it darker. I'd probably leave the inside of the drawer unstained in this case just to avoid any risk of residual odor if you're using an oil stain.
Second choice would be gold or amber shellac which will also give it some color and a nice protective finish. Mostly second choice because it'll be tricky getting the shellac into the insides of that little drawer without gooping up the sides. You can also put this over an oil stain just fine, ditto on probably leaving the inside unstained.
You could also leave the inside of the drawer mostly/entirely unfinished and do the outside in whatever..
