Extremely large ziplock back (2-1/2 gallon size) with the bird and brine in the cooler. Or in the pot, inside the fridge.
I wouldn't place it right into your brew bucket since I wouldn't 100% trust it to not get something nasty latching on. Especially if you've already used the bucket for any batches. If you're going to use a bucket, get a new one and use it only for that.
I use either Lexan containers, or food safe/grade plastic buckets to brine in. The Lexan is a very hard plastic, that won't pick up anything. Plus, it's very clear.
Still, when I'm using a brine, I tend to place everything into a ziplock bag before it goes into the container. Besides having less to clean, or worry about, you use less brine to do the job.