It's so hard when you can't just make it go away.
My DC started off so well (probably too well with hindsight) and then a few months in developed huge issues with injecting etc, tears and tantrums etc.
We tried to help by taking over the practical difficulties where we could but ultimately it wasn't until DC moved to a pump, Libre and monitoring more often using apps and other tech that things got a bit better and the regime seemed a bit less obvious and traumatic for him.
I don't know if your daughter is doing manual injections daily or not but a pump If that was an option may help with that side of things if they are a struggle?
Tech continues to improve at a rapid rate and I predict that not too far in the distant future Type 1 diabetics will be able to have tech on them that will do everything with very little manual input required. That may give her a glimmer of hope.
I think the kick back and upset is a natural reaction to the realisation that that the condition isn't just going to go away. We still have days where the injustice is hard to manage but generally there's bounceback when days go well.
In the absence of being willing to speak to a psychologist all I think you can do is to keep talking and take on the burden of the testing and treating regime where you can.
We also had an approach of business as usual so despite it being very scary and a logistical nightmare at times to send DC off without us we encouraged life/hobbies/outing/camps away etc to continue as before even if it meant huge preparation for us (that he was fairly unaware of) in advance. I think emphasising that normal life can go on is really important as is reassurance that they are no different from the next person and enabling them to be just like their friends (albeit with a lot more forward planning).
Another thing that might be worth doing is using young role models who have great lives despite living with Type 1 - James Norton, Nick Jonas, Amelia Lily etc if you think she's buy into that?
It's such a tricky situation 