I do agree if the bird is hypothermic and quite ill, she should not be subject to water. If she does indeed have a stuck egg in this situation, if the egg is at all accessable, the egg should be broken and removed.
Just yesterday I helped out this person with a quail and a stuck egg. Read here....https://www.backyardchickens.com/t/950956/hello-everyone
In post number 27 the bird is now back in the flock....https://www.backyardchickens.com/t/950956/hello-everyone/20#post_14736402
It is very easy to do. Put on a latex glove with some KY Jelly or Vaseline on the index finger and gently insert your finger into the vent going straight back. If she has a stuck egg, you will feel it within the first inch or so. If you feel nothing or can go down with your finger, you are in...
I have had great luck with soaking birds in warm water. As long as the water is not so warm it burns them, warm water helps relax the abdominal muscles which can be holding the egg in. When they have been pushing and pushing, the muscles in the abdomen tighten up and holds everything in. If the...