Could I offer a slightly different perspective? With absolutely zero judgement, I promise. I'm an anorexic, I also have severe clinical anxiety and depression, PTSD due to SA and rape, and not the most "stable" of people.
An eating disorder helped me deal with the trauma and confusion of my life, on and off, since aged 11. I understand. I get it. Eating disorders come in all forms, no one is more noble than the other. So I truly "get" your BED. And I hope you will beat it.
However, over the three decades I have suffered with anorexia to various degrees (at my best, and now, you wouldn't even know, I just look like a size 6/8 slight control freak) sometimes I overcompensate and order the cheeseburger, other times I cancel social plans as the thought of eating out makes me want to curl up in bed with carrot sticks and starve myself.
I've been to the doctors with numerous ailments over the years, including mental health issues and suicide ideation. Every time, I've been told to "eat well and put some weight on, it will help" on the flip side to "just lose some weight"
there's been times I've been so offended "I'm not even thin, I'm as big as I've ever been, I'm certainly not going to put weight on for some BMI calculation"
Other times I've admitted I have been restricting, or calorie counting, and said I understand I need to be more flexible.
Then other times I've broken down in tears because I KNOW I'm a recovering anorexic, I KNOW I need help when I get too thin, I KNOW I can't expect to sleep 6 hours, work a 12 hour shift at a manual job and be full of energy.
I guess what I'm saying (very ramblingly!) is that sometimes the doctor is right, and sometimes the doctor is wrong, but honestly a healthy baseline is always best. I was NOT healthy at 6.5 stone despite having a full time job and 3 "meals" a day. Similarly, my zinc deficiency was down to just that.
All I mean, in the gentlest possible way is: losing a bit of weight can't hurt. Same as for me putting on an extra half a stone really helped with other things in my health.
It's all a spectrum, give your body the best possible chance. You losing/me gaining those extra pounds could only help our respective overall health journeys, it won't hurt our chances. I'm now over 7 stone, I feel I look disgusting but I actually do feel healthier and with more energy than at 6.5 stone. Maybe try that with half a stone to start with from your end.
I wish you nothing but the best, I hope this message is received in the spirit it was intended, without judgement, and with care and understanding. And love.