Several options:
- As Dave says, brine in bag and put in fridge (without container) if you have room. Unfortunately EVERYTHING leaks when I try something like this so it wouldn't be my preferred method. Luckily, not everyone is as clumsy as I am.
- Brine bird in bag, then put bag in cooler with enough ice to maintain safe temp.
- Spatch or cut bird into parts (wing/breast & leg/thigh) before brining. Should fit in container in fridge that way. Use ribs, back, etc. for stock.