Apologies for the length of this....
Doctors and nurses are notoriously ignorant of Binge Eating Disorder.
(Its not necessarily their fault , they have so much to remember and its only in recent years that the seriousness of it has been realised)
OP = you say you have tried and failed to stop binge eating, please understand that it is a poorly understood severe mental illness and know that you are neither alone nor at fault. Blaming yourself for not stopping binging is like an anorexic blaming themself for not being able to eat.
Binge eating disorder IS a classified eating disorder alongside anorexia, bulimia and non specified eating disorder.
It is just as serious and like the other eating disorders puts your health and life at serious risk ( I say this not so much for the op but for anyone reading that may think that it is not a serious life threatening disorder,)
For those with Binge eating disorder there is not as much help out there as for other eating disorders (and there is not much help for them either) and very sadly to a certain extent, unless you can find a sympathetic practitioner you have to self advocate in order to find help.
The reason I say that is that before complaining, I would return to gp and have your reasons written down so that you can discuss things calmly as to a certain extent you want to be able to work with gp.
If that doesnt work then you can then explore other avenues. Either that or if possible see another gp. You may want to push for an official diagnosis in order to access help. I understand that this is not ideal but it is sadly the case for many things in the uk that you have fight to get the resources and help that you need.
I do not think you are being unreasonable to want a different drug that is less likely to cause weight gain. I think your fears are justified.
ETA
Ive just read on before posting and have noticed you have written you do not think you have an eating disorder. Op whilst none of us online can diagnose, you yourself have stated you have binge eaten enough to put on 5 stone, its affecting you mentally and physically and you also used to purge after eating.
I would advise you as others have to visit
www.b-eat.co.uk/about-eating-disorders/types-of-eating-disorder/binge-eating-disorder
(they also have a helpline you can call ^^)
There are also some long running threads on mumsnet about binge eating in the health/eating disorder section.
www.mumsnet.com/Talk/eating_disorders/2652882-Binge-Eating-Disorder-Support-4
Please seek help op and do not try to struggle on alone. If the gp isnt helping, change them, find support groups, online forums, there are multiple youtube videos of peoples own stories of dealing with it, educate yourself on the different view on treatment for binge eating and find what best suits you.
There are different views on how to treat binge eating disorder, ie for example those that say that you need counselling and anti depressants to solve the root cause and those that say you do not , ie that it is a severe habit that has developed as a coping mechanism and the root causes do not have to be solved .
There is much better access to services in America and there have also been success in america treating it with medication ie Topomax & Vyvanse although i do not know if they are licensed to be used in the UK.