I looked up the BMI for 120 @ 5ft6 and its in the lower range of "normal". I just wouldn't go below the healthy BMI and you should be fine. People love to comment on each others weight. Look at Britney Spears for example when she gains a little weight and gets pudgy everyone starts calling her "fat". If she loses weight everyone starts saying shes "too skinny". Just do whats comfortable/healthy for you. And dont worry what everyone else thinks. Its not their business.

I'm also your height and I weigh almost 20 lbs more than you and you can certainly see my bones. I go round and round and round whether losing another 5-15lbs is a good idea for me. People often think I weight a good 10lbs less than I do and personally I would never venture below 135, because of this reason, but that's me.

If you're seeing an ED specialist to determine why you want to lose weight then I'd certainly take a pause in your weight loss and maintain for a bit. I've been doing this myself just to assess and determine what my real goal is. I think I'm starting to realize that in the end, in the big picture, it just really doesn't matter. As long as I'm happy, healthy and feel good about myself, the number on that scale just really doesn't matter.

Might already be fine
 

I honestly think part of the issue with people thinking others way less than they do is how many people "fudge" what their weight actually is in conversations. I have gotten the same thing. When I weigh 180 people think I weigh 150 because I'm so tall. When I tell people I weigh 196 they don't believe me. It could be a similar sort of thing here, but you may be pushing yourself too far in your weight goals as well. One weight doesn't work for everyone and you have accomplished a lot.

I think you are taking the proper steps by seeing an ed therapist. Remember weight is just a number and does not define you. It's just one part of you. You are you not matter the weight!