First of all, ((((((((((HUGS))))))))))
I know how hard it is when you are so desperately in need of help with your child and you have to send them to a relative for a while just so the rest of the family can recover for a bit. It really is hard and it breaks your heart. I had to do this with my oldest son at times and when he was 14, I had to accept that it was just too dangerous to have him live in my home and I had to have him go live with my parents. For us this was a great thing and now he is 25. We have a very loving relationship and he is an amazing young man. So there is hope and it truly can get better.
Please know that she falls apart at home because she knows that you will still love her. She holds it together at school and acts well because she thinks/feels/knows that she would be ostracized/bullied if she acted out and her life would be so very hard. But it takes all of her energy and focus and everything in her to keep it together and when she gets home she just literally falls apart. She simply cannot cope with anything. She knows that you love her with every fiber of your being, so she allows herself to just lose it.
As a mom this is HARD. We get ALLLLLLLLLLLLLLLL the ugly behavior because we give the unconditional love. It means we are doing things right and it totally HOOVERS!
It is time to figure out how to help your daughter develop better coping skills. My son's sensory issues were so bad that he was 9 or 10 before he was able to go to school for the full week. I remember looking at his report card and being so incredibly PROUD that he only missed 1/4 of the days in the 9 week period just the year before, and then suddenly he was only missing 2 days of school in an entire year. It took me insisting for him to stay home those days - he had a fever and was throwing up but he still wanted to go to school! Up to that point, if I sent him to school on a bad sensory day, he just shut down. He would sit there and sort of vibrate, shake back and forth like he was cold only he wasn't. He just couldn't cope with the noise, the lights, the smells, the anything. He would chew on his shirts until the entire fronts were soaked through. I think he was probably 10 before I bought him shirts anywhere but garage sales, or paid more than 50 cents for a shirt because he was just going to chew holes in them.
My son didn't react violently to sensory input, but I know kids who did. Their parents had to find ways to provide the sensory diets they needed so that they could learn to self calm without reacting with rage. I personally have bad sensory issues and they get worse as I get older. The sound of someone chewing, especially with their mouth open, is a major problem for me. Not only does it make me throw up, it sends me into a rage. Listening to a football game also makes me very angry. I have no idea why. I don't like football and it makes no sense to me, but just having it on in the background at a low volume shouldn't induce a huge rage. But it does. So I avoid it like the plague. I especially avoided it when my kids were little because I didn't want to react with rage to them when it wasn't their fault.
Your daughter needs to learn to manage her sensory issues with a sensory diet. I don't know if you have read "The Out of Sync Child" by Kranowitz, but it may be a huge help in understanding your daughter. I strongly recommend getting a copy of it, and also a copy of "The Out of Sync Child Has Fun" by the same author. If you have to get one from the library and buy one, buy the Has Fun book. It is PACKED with activities that provide the sensory input that will help various types of sensory needs and it is also seriously fun. It contains tons of ways to make the activities less expensive and that is always a benefit. The activities will help draw the entire family together - they are fun for everyone, not just your daughter. We actually wore out several copies of this book because my kids loved the activities so much. Every kid in the neighborhood (several neighborhoods as we moved a couple of times) would beg to do activities from the book! Not all the activities wind up the kids, some help with calming. It just depends on what stimuli the child needs.
I don't know if anyone has brought up brushing therapy with your daughter, sometimes called the WIlbarger Technique ? This is taught to you by an Occupational Therapist and is used to help the brain learn to use sensory input. A soft brush, often a surgical scrub brush, is used to brush the body in a certain pattern and then gentle joint compression is used. This actually helps rewire how the brain uses sensory input from the various parts of the body. It has to be taught by an Occupational Therapist (OT) as if it is done over certain parts of the body it can have bad side effects. But otherwise it is very safe, very gentle and very effective. Most children find it pleasant. When we were taught it for use with my youngest, I was told that it worked on young children but not on older ones. I think your daughter would be young enough, given what I was told. But this was ten years ago. I disregarded the age thing and found that it helped my older children quite a bit also. (My entire family has significant sensory issues, we just figured we were strange.) In fact, my daughter was in the 'helper' stage and insisted on brushing Mommy, which the Occupational Therapist (OT) thought was cute and harmless. It was harmless, in fact I found that it actually helped quite a lot. I found that MY sensory issues actually improved.
I would recommend this but I recommend using a private Occupational Therapist (OT). The school Occupational Therapist (OT) only cares about what impacts behavior at school. A private Occupational Therapist (OT) is far more comprehensive and looks at the child's entire life and what impacts all of her behavior and life. So the treatment plan is vastly more complete and helpful in my opinion and experience. If you get an Occupational Therapist (OT) that isn't this way, find another one.
I do TOTALLY agree with changing your daughter's IEP because something at school has her completely stressed out or she wouldn't be falling apart at home. You can get an educational advocate by contacting the state department of education. This advocate is FREE to you and is a HUGE help with IEPs and anything else with the school.
None of this is your fault. Your daughter is wired differently. Her brain works differently. She needs to be taught how to handle this and it can be tough at times. But that is NOT your fault. You have to balance her needs with your son's needs and your own needs. Those are all EQUALLY important. Your daughter's needs are not more important than anyone else's, and that can be easy to lose sight of during times of crisis.
It really isn't your fault.