I know from an individual with T1 basis how trying to find an endo can be demoralizing. I can relate to the whole weight issue in general. In days gone by, like the stone age, if I weren't T1, my particular metabolism would be a distinct advantage, especially during a famine. Now? In a period where I'm in good, stable control, I don't feel hunger or need that much in terms of calories.
When I exercise, however, the amount of intake in terms of calories has to go up, enough to remain stable but not enough to throw highs into the Requirements of Existence. Insane hunger, for me, comes with instability in levels (or recovery from instability). When I reach stability again (and it's easier me...I'm 33 years old), the hunger and eating goes away. It's like chains falling off. With high blood sugars all of the time, or a severe shock of some kind, all manner of interesting and frustrating metabolic situations happen.
Out of control, I am controlled by hunger. Also, T1 interferes with sleep, which in turn interferes with metabolism...I don't know what it is like to be a parent of a T1 at all. I know what it's like to choose between gas and food. I also know what it feels like as one person to have inadequate or even non existent medical care. I know what it's like to not have insulin.
I'm sorry that you are going through something that sounds much worse. I hope it gets better soon. I can't say anything more than that because when it comes to your experience, I am completely ignorant. You're a T1 parent, however. In my eyes, you are a hero.