My heart goes out to you. This is one of the most difficult things about keeping chickens. Deciding when to euthanize. (Culling to me is what people do when they keep a breeding flock and weed out the poor genes.)
Here's how I make the decision. In fact, I thought I'd be making the same decision this week that you're facing. I have a five-year old Cream Legbar hen that came down sick a few days ago. She wasn't eating or drinking and she was very lethargic. I wasn't sure what was wrong so I treated her for egg binding and bacterial infection and, just recently, decided she might have coccidiosis, so I added Corid to her treatment along with an antibiotic.
I gave the antibiotic and Corid treatments time to work, after ruling out egg binding. There was no real improvement for the first few days, but she wasn't getting worse. It wasn't yet time to make a decision. I would give her more time, watching to see if she took a turn for the worse. She held on, and today for the first time, she is eating and drinking. She's improving. As long as I see progress and that she's choosing to fight to survive, I'm happy to keep caring for her.
So, this is how I make the decision. If a sick chicken is not improving after a reasonable length of treatment, And if a chicken gets worse, and keeps getting worse, I don't let them continue to suffer. I euthanize.
Under my criteria, if it were my baby chick, I would decide to euthanize at this point. I hope I've helped you make a very difficult decision.