Mia,
This problem that Sugar is having is a soft tissue thing, nothing would show up on an X-ray. I don't even think they would biopsy to be honest. This is an abscess of some sort, for whatever reason. It has obviously been there a while as it has been growing. Edema (the swelling) she has experienced may be part of why she is breathing raspy. With that much swelling, there is a lot of pressure on her head and what is in her head. You did great by relieving some of that pressure, but the swelling going down is a slow process and if you keep treating the infection inside that tissue, it will eventually go down. You are doing about as much as you possibly can without vet intervention. Get it cleaned, keep it cleaned, treat with antibiotics.
I had a cyst in my back 15 years ago, and it needed to heal from the inside like your hen's is going to need to do. I had to have my husband pack this gauze stuff in it, it had to be changed every day, so that it would heal from the inside out and not get germs in that wound infect it as it did heal. Perhaps you could get something like that to help Sugar heal once the dead tissue/blood/pus is all cleaned out? I would ask a vet at the very least.
I know you are getting lots of advice from everyone on here, so it may be hard to keep up with or decide who is right and who is wrong. You sound like a very bright young lady and you are doing wonderful work with this bird. If she means as much as you have made obvious here, then keep working at it and you will get her back to her old self soon. Already she sounds like she is doing way better than when you made the first post. She is responsive and has eaten, and that is saying something.
I do understand you parents about the money, and sometimes it is not about the money, I know. You can do a lot without a vet when it comes to chickens, so I hope your efforts find their reward in you and Sugar spending a long time together.
Best wishes and big hugs for you
Theri