OP this sounds like an awful situation, I can't imagine how worried and stressed you are. As a mum, it must be heartbreaking.
In your original post you asked for advice, so I'm going to try share a few things that might be helpful. They definitely aren't criticisms, just maybe some things to help you navigate this situation. I'm a mental health nurse and have worked in a camhs setting, but only for a short time. Others may be able to offer more help based on their experience
The first thing to say is, you are highly, highly unlikely to get a diagnosis of anything for your daughter. It is generally not deemed helpful with young people and often it ends up being a stigmatising badge that young people carry around for years. If you can psychologically drop a diagnosis from your list of wants in this situation, it may be helpful for you.
I know you feel that medication might be a last resort and I see why that might be. Meds are hit and Miss with young people and often really increase risk (suicidal urges can increase). Meds on their own definitely won't fix whatever is going on.
There are a couple of important factors here, the first is trying to understand what is driving these behaviours, and the second is to find effective ways of coping with the distress she is experiencing. She does need coping skills, but there is no one size fits all here. I wonder, have any of the professionals who have offered coping skills taught you any of them? Sometimes the skill is useful but if the young person has become too distressed, they can go passed the point of being able to remember/have motivation to use them effectively. Sometimes having someone to coach the person experiencing distress in applying the skill can be really helpful, and it might also make you feel a little more empowered. I am not sure if they have gone through any of the DBT distress tolerance skills with your daughter, but if not you can find great videos on YouTube of how to do them, often taught by young people who have used them when distressed themselves
It might also help you to know that you aren't being fobbed off with "just a nurse". Mental health nurses are specialists in their own right and make these kind of assessments every day. They should be taking about your daughter in their multidisciplinary team meeting and discussing her care. This nurse won't be working with your daughter in isolation. Having said that, if you feel you aren't been heard, then sometimes documenting your concerns in writing, can be reassuring for you and also reduce likelihood of misunderstandings. I would compose them in an email (as factually as possible) and send them to the team, asking for confirmation that they have been read. It also ensures there is a paper trail of your concerns. The nurse will also have a team manager (likely to be another nurse), who you can ask to speak to to have your concerns heard. If you feel like this still hasn't helped, there will be a formal complaint procedure for the trust which provides the service.
Finally, have you looked at young peoples crisis services in your area? Sometimes there are particular suites on offer during the evening as an alternative to A and E, with crisis professionals on hand. Beyond that. if your daughter isn't feeling safe, take her to A and E. every time. It is a place of safety and sometimes being a repeat attender can trigger heightened care. The other thing to encourage is that when she sees the staff, she shares with them what she is sharing with you. Sometimes young people do say different things and that can impact on the assessment of risk that is made.
Sending you lots of good wishes as you navigate this awful situation. Your daughter is lucky to have you as an advocate