Last year I did 3-4 birds at a time in my large cooler. I poured over a more concentrated brine based on the one in the recipe section from poohbah and ice. Left it set over night and it was AWESOME! Making enough brine was a challenge and took my large pot a couple batches to make enough for all the turkeys but it worked out. The solution for this one was sorta on the expensive side since it had a ton of fruit juices and a "tea" to make from boiling fruits. Try it out and you wont be disappointed!
But like everyone said, you will either get a nicely brined bird followed by consecutively less brined birds or you will get a batch of sick birds with the way you are proposing. I'd just do'em all at once. Will your pit accommodate multiple birds?
Cooking for the love of it.