I think I am finally getting it worked out. First though, I have to stress that this isn't messy pooping and the issue isn't the poop at all, it's the urates and urine that drip down, get her feathers all wet, the feathers eventually fell off due being wet with urine all the time, and now she just has inflamed bare skin below her vent that is always getting wet with urine, plus white urates crusting around her vent.
A few weeks ago I finally took her to a different vet because I just can't handle constant worry, especially with winter coming. This vet has been so much better. He looked analyzed her poop and identified two kinds of bacteria and two kinds of fungus in it. I have no idea why the first vet didn't do this, it is quick and not expensive. He put her on two new antibiotics to fight the two kinds of bacteria. He thought that it was possible that the fungi were there because of the bacteria and that once the bacteria were gone they might go away too. We were hoping for this because to figure out which fungicides were needed is expensive; plus, many have some serious side effects. I thought maybe she was getting a little better for a while but then it seemed that she was the same so I took her back again a few days ago. He tested her poop again. This time, almost all of the fungus was gone - fantastic news. Also both of the kinds of bacteria that we were treating her for were gone, but there were two new types. He didn't think that was so unusual - the new bacteria moved in to the opening created when we got rid of the first two kinds of bacteria. Now she is on two new antibiotics for the two new bacteria. We also tested stools samples from my other two hens and one had the same two types of bacteria (and so she's on the antibiotics now too) and one had no bacteria at all. It has only been a few days since the hen I was so worried about has started the new antibiotics but her symptoms seem a bit better and I'm feeling really hopeful.