What's the situation with your doctors? Have they known you since you fit the AN criteria, and are now assuming that weigh gain is good, and that you don't actually have a problem with binging? The professional assumption with AN b/p subtype is that they're subjective binges, rather than "real" binges. You're probably familiar with the idea of an anorexic calling it a binge when they eat what anybody else (including truly healthy people) would consider a normal meal, but it's considered a subjective binge simply because the anorectic feels out of control. The whole "OMG, I ate a half cup of plain yogurt, a peach, and a and a peanut butter cup! I'm going to gain eighty pounds!" thing. If you're at an anorexic weight, they assume that's what you mean by "binge," no matter what. Doesn't matter what you say, doesn't matter what your parents say, because "purging doesn't work" so it's supposedly impossible that you're when you say "binge" you mean anything more than 500 calories. Yes, anybody who lives in reality knows damned well what crap that is, but somehow ED professionals maintain the delusion.
Wellbutrin is generally considered to be contraindicated in bulimics because it lowers the seizure threshold. The risk seems to be debatable, but that would at least be fairly valid reasoning for not prescribing it to a bulimic. Not prescribing a medication to a bulimic because it decreases appetite, however, indicates that the professional is working off of myths rather than medical evidence. Unless the professional believes that you would misuse it, taking it specifically for appetite suppression in a way that would cause malnutrition, that aspect is the least of anybody's valid concerns.
I really, really wish that eating disorder professionals would learn to make a reasonable, useful distinction between anorexics and bulimia, but that doesn't seem likely anytime in the foreseeable future.
FWIW, I have a long history of struggling with bulimia; I spent my 15th birthday in Renfrew Philly, just over 15 years ago. The most helpful medication combination I've been is Wellbutrin and Adderall. (I also have fairly severe ADD, but also have suspicions that that's not unusual among bulimics, and most of what I've read reinforces that impression.)
I have not been underweight on that combination, although part of the reason for that is the fact that I'm afraid it would be a problem with staying on the medication that actually helps. My weight has been fairly up-and-down during that time, so I've both lost weight and gained weight while on both medications. It does make it easier, though, especially in being able to identify hunger and fullness (which seems to be an extremely common problem in people with AD(H)D.)
If your doctor is hesitant about prescribing Wellbutrin based on the possibility of appetite suppression and not because of the seizure risk, that's a red flag to me.
And truly, having a fairly thin female psychiatrist helps. Men and people who are overweight both have issues understanding the problem. A thin woman who doesn't have an eating disorder may not truly understand, but at least it's not a completely foreign concern.
It can be really hard to properly articulate the issues that are most difficult for you, and even more difficult to find somebody who will even attempt to understand where it's all coming from and how to deal with it. In my experience, ED professionals can be the worst. They're more familiar with ED problems medically, but sometimes that's a problem. Unless you're the stereotype of a disorder, a doctor specializing it is less likely to bother to think or understand what your own issue is. They end up just mashing you into the stereotype. It's better to see somebody who doesn't know than to see somebody who "knows" something false.
Is your doctor well-informed?