Bathing a chicken is actually surprisingly easy. If I'd realized it would be this quick I'd've done it earlier. They were not precisely cooperative, but I would characterize their responses as puzzled, perhaps offended, rather than panicked or angry, so one hand on top to minimize flapping and one hand in the bucket agitating the water underneath was quite sufficient. The internet advised me to use very warm water and I think that helped.
We'll see how they look when they dry out, and whether they can keep that area clean on their own now or whether we'll need to repeat the procedure in the future. But for now I am quite satisfied with the chicken bathing experiment.