Will there be air movement in the box? Changing the temperature of stagnant air is difficult, even a small circulation has a big effect (think ceiling fans). PJ is at top, relying on convection in the 3x3x3" space.
Will the PJ be mounted on the top or the bottom of the box? Top would be best, cold air sinks. No sense in cooling cold air. --- YES
How fast do you need to cool worst case (from 35C to 0C)? -- No rush, 2-4 hours
Peltier units work by creating a temperature difference between to the two sides. The cold side will be the box side, the warm side will be mounted to a BIG ol' heatsink. The heatsink will act as an 'anchor' for the temperature difference. If the heatsink temperature changes, the cold side will move (almost) in direct correlation.
You are aware that peltier junctions are power hogs. A quick search found a 40mm X 40mm 72W unit that requires 6A at 12V ...That's ok, compact size and convenience matter more, here .