You do have a dilemma. I don't know how many chickens you've had to try to nurse back to health, but I've had my share. The most difficult aspect is deciding when it's not in either of your best interests to keep on trying.
I've developed a sort of formula. Whether sick or injured, I give my all to the endeavor as long as the patient is showing some progress or has reached a point where she's able to survive on her own despite a chronic condition. That means she must be able to eat on her own, get around enough to hold her own in the flock, and most of all, demonstrate she's getting some benefit from being alive.
For example, I had a rescue hen I adopted after her flock was all killed by a predator. After being with the flock for several years, I noticed she was behaving withdrawn, sitting on a cushion all day. I looked her over, finding a huge, scabby abscess on her tail base. For the next couple weeks I debrided and cared for the wound and gave her an antibiotic. Nothing worked, and the wound refused to heal. Along with that, the other chickens were noticing this one was not feeling and acting well, and they began to pick on her. One day, unknown to me, she was hiding from the others on the other side of a wood pile where I was working. A accidentally knocked a log off and it fell on this poor girl's wounded tail, and the screeching that followed finally decided things. Life for this chicken was no longer to her benefit. Since I euthanized her, I've decided she likely had squamous cell cancer on her tail.
On the flip side, I have a ten-year old lame hen named Su-su who can get around, but with great difficulty. But Su-su has a full life and not only feeds herself but she took on the job of raising chicks which she is still currently doing.
https://www.backyardchickens.com/threads/we-know-broody-hens-how-about-a-nanny-hen.1407935/ I had tried to treat Su-su's lameness at one point, but nothing worked. But she seemed to always have a "purpose" and was able to hold her own in the flock. My oldest is twelve and blind, and she also holds her own, manages fine competing for food, and she never backs down when some youngster challenges her.
To sum it up, as long as a chicken can still live life as a chicken without more than a bare minimum of assistance from me, they are allowed to live their life, but when it's obvious that life is a struggle for them and for me, with no positive benefits for either of us, I end it.