Generally, WW rangers assign camps based on when you are ready for check-in, the size of your group, and the size of the other groups that intend on doing an overnight trip. But it's rare for folks to not be able to get something that works for them.
In your case, either Bald Eagle or Big Horn would suit you. Big Hole also would work theoretically, but it would be a chore to get all your stuff up to the grassy saddle where there is lots of room.
Talk to the ramp ranger when you arrive, and tell her/him what you have in mind so they know how to try to fit you into the overall picture for that day. Then get your gear ready and finish the check-in process as soon as you can, so they can make a final assignment.