I would check around for hours, ask who can meet you at their locker after hours, ask what is the latest they will meet you. If you stick one late evening and it is 10 or 11 pm, will they meet you to get the carcass in the walk in? If you cannot hang it at home overnight then load up the cavity with ice from stem to stern and lay it so water can drip out. We would lay the deer down on a tarp, load it with ice and fold the tarp over. Transport with ice and remove before taking in.
Most places don't give a rip about taking the time to hang a deer- especially if tbey are an inspected facility and have beef and pork to cut. Your deer might lay in the cooler on a suspect floor for several hours before they get to it. They cannot butcher your deer while they have inspected carcasses to do. I would take a cooler, cutting boards, knives, lanterns,etc.. along with me. Skin and debone or quarter your deer, place meat in trash bags and lay on ice. Moisture can cause big problems if meat is above 45°. Anything you are going to turn to burger does not need aged. Backstraps and filets generally do not need aged but can be held in fridge for a few days if you wish. Anything you plan to steak or keep in roasts can be aged in fridge for several days and not take up too much space or keep iced in cooler as Lycan suggested. We got to the point where we could skin and bone out a hanging deer in about 90 minutes or less.