:hugs I'm sorry things are not going so well.
Is vet care an option?
You may need to consider antibiotics in addition to cleaning all the pus out.
I agree with you in considering her overall health/long term recovery too. Sometimes we have to make hard decisions.
You are doing good!
O.k. that yellow cheesy stuff - that is chicken pus. You will want to work on pressing that out. Take your time, you may have to work on that for a couple of days. Flush well in between with saline, betadine or Hibiclens. See if there may be some impacted feathers in...
Looks like it could be a feather cyst, a festering injury or abscess.
I would clean it well with Chlorhexidine (hibiclens) and investigate it further to see if there's pus or curled up feather(s) underneath that.
https://www.beautyofbirds.com/feathercysts.html...