First, a proper drain would have built-in trap to block air transfer to and from the box from the outside. (The water functions as a barrier (not perfect, but at least managable.)

Secondly, What's the path of the water from the "hidden" hose outlet? (There are inspection devices that exist that plumbers and others use to see around corners, etc.)

Thirdly, if it were me, I'd consider

o Filling the existing drain (glassing it over, etc.);
o Putting in a new drain where you can get to it. You'll need to so it so you can insulate where you make the penetration. If it's not the pure "bottom" of the box, then put a plexiglass plate above it so you can create a sump and keep your food dry, etc.;
o Put a valve in that drain line, insulate it if the line leaves too much length from the cold zone to the valve; and,
o Then put a hand pump on in so you pump it into the sink when you want to empty the water out. You might leave the pump there and just have hose coiled to lead to sink, you might pull the pump out eacy time and connect it.