Well, I asked two vets (my mom and my sister-in-law) and I will share what they said, but I am still very open to others' input, as neither my mom nor sister-in-law are avian vets and I am very open to hear other pigeon owners' input.
First, my mom wondered whether whether her vent was egg bound. She also said the oozy stuff looked to her to be puss. She recommended I palpate it to see if there may be an egg. My sister-in-law, who is not an avian vet either, but owns chickens, said that egg bound birds do not present that way so she thinks the bird is not likely egg bound. To boot, I'm not even 100% sure she is a female, and I am not aware she has any mate at all, so I find it unlikely she has an egg.
My sister-in-law did some research and said it looks like it could be a vent/cloacal infection, and probably not a prolapse. She recommended I first soak it in a saline bath, then flush it out gently and keep an eye on it. She recommended that I run my loft with a dose of anti-biotics just in case, especially since I got all my birds from a large loft elsewhere when they were squeakers, and they have never once been treated with anti-biotics or de-wormers. I also loft fly them quite a bit, so it is a decent preventative measure in any event (lots of ferals around). I am, however, aware that some fanciers avoid administering preventative anti-biotics.
