I'm so sorry. Things like this, no matter the actual diagnosis, seem to be ~extremely~ difficult to treat. Good for you making the decision and not letting this little one suffer :hugs
https://www.merckvetmanual.com/poultry/omphalitis/omphalitis-in-poultry#:~:text=(Navel%20Ill%2C%20%E2%80%9CMushy%20Chick,%E2%80%9D%20Disease%2C%20Yolk%20Sac%20Infection)
Just in case, here's some reading on what my chick had. I purchased her at a little over a week old. I believe she was around...
Was the skin red like that before? Is it feeling any different to the touch? Honestly and unfortunately, it looks just like the chick I mentioned that I had a while back. Still, I hope I'm wrong here
Unfortunately it died as a chick. It was a few weeks old, because I didn't purchase the chick until it was a little over a week old. The chick acted fine, it just had this odd looking and feeling abdomen. I believe mushy chick is extremely difficult to treat. I really hope that's not the case...
The one time I had a chick like this, the abdomen felt odd. The best word I could find to describe it was that it felt heavy and full. It felt off and looked off. I believe in my case it was mushy chick disease.
Maybe it's just me but the abdomen looks a bit different than it's flockmates. The vent doesn't appear to be prolapsed, in my opinion. How does the area between the legs feel, compared to the others? Is it possible that it's a bantam and the others large fowl?
Edited to add: how does the naval...