Rather than having to worry about brining, just buy a brined bird. Most commercial birds are brined anyway, the label will say injected with a solution of xx%........For a holiday once a year, Thanksgiving is worth spending the extra $$ for a Butterball that has butter injected also.
That gets your problem down to storage. Use the small fridge for storing some of the "other stuff" that's smaller and will work in the dorm fridge, put the big bird in the big fridge.
Cooler will work too, just make sure the bird stays below 40*.