In a way, yes. More generally, we're asking about the limits of self-determinism, body integrity rights, and mental fitness. If we refuse to cut off a man's leg, we're essentially saying "you are not mentally fit to make this decision."
I'm very much a believer that people should be free to do whatever they want so long as it doesn't harm anyone else. The trouble I run into is defining "harm to someone else."
For instance, in Wisconsin, we have two completely at odds laws. On one hand, it is illegal to drive without a seatbelt, on the other hand it is perfectly legal to ride a motorcycle without a helmet. I tend to side with the people who want to be able to choose (poorly) to drive without a seatbelt. I would never make that choice, but I don't think it should be codified in law.
However, if that person drives without a seatbelt, crashes and gets hurt, who is really being harmed. The person driving, obviously, but it was their choice, so "oh well". But then, if they have family, that family is distraught, not to mention perhaps financially dependent on them (or even just paying for their medical bills). Even if the person is completely familyless, they're now in surgery in the ER, taking the spot of someone else who may need surgery but not quite as urgently, or who merely got injured a little bit after the first person.
In the case of BIID, I'd like to agree that if they truly want to amputate an appendage, and are mentally competent enough to make that decision, then let them. But then again, if they have a family who depends on them for financial support, or if they'll be thrown onto disability and now taking resources from all tax payers (or others who could use the assistance) based on a choice rather than an accident, I'm less inclined to their side.
I guess I'd say, if they can be treated and brought closer to the "mainstream" then that should be the first choice.
Would that mean that if a person wants to have an amputation as a form of extreme body modification, fully understanding that the procedure is not in any way necessary, that the limb in question is not harmful, that the procedure carries risk in and of itself, and the consequences afterward, that we should allow that person to receive the amputation? With no delusional belief motivating the desire for amputation, does the argument of mental fitness no longer apply?
The thing to look at is again, harm to others. Getting piercings, tattoos, scars, etc doesn't stop a person from being able to work. There may be some positions that won't allow a lot of tattoos or piercings, but that's actually becoming more and more acceptable in many work environments. What removing a part of your body does is remove certain occupations from being a consideration. Without arms, you can't carry things, without legs, you can't do many jobs, etc.
It might come down to how "extreme" the removal will be. Removing a pinky probably won't have much of an impact, and if ti will really improve their quality of life, then I say "Off with the finger." If they want to remove their leg or their arm, we need to see what the plan is for after the surgery. If they have a job that doesn't require that appendage, and they'll be able to continue to take car eof their obligations, then again, if it will actually increase their enjoyment of life, let them.
Sorry if this was a bit of a ramble, but I'm trying to figure this out myself.