I'd say you DH needs to say something to Bm ( i know its hard to get done sometimes but yest this is effecting his child and BM needs to get over herself) sadly all you can do is be there for SD and she will come around.

I was actually going to suggest the signal thing. I think that is a great idea. I would tell her that you guys can do the signal when BM is around and not to worry about hurting your feelings otherwise, because you understand. I wouldn't tell her not talking to you has to be her choice, that is the situation she is already dealing with. If she feels bad, she does want to say hi to you but doesn't feel like she can. Just tell her it is okay, you understand and when she is ready, she can stand up to her mom.

Don't put yourself in a position where SD has to directly choose between you and her mom. Disengage yourself from BM. Don't go to drop off/pick up/ sports games /school events where she will be.

Then at the same time talk to SD and tell her that you and her mom aren't friends and that that is ok not everyone has to be friends. But that doesn't mean that you and SD can't talk anymore or that anything needs to change with you two.

I really don't think that a signal is a good idea. The girl is already getting into trouble for talking to you. Imagine what will happen when BM finds out about your secret signal, she will be furious. I don't think that you should be encouraging your SD to keep secrets from her mother. Just take yourself out of the picture and don't go anywhere where BM will be, or alternatively just tell SD that it is ok with you if she doesn't want to talk when she is with BM, that you understand that it is awkward for her.

Order "Welcome back, Pluto" and have her watch it with you. We bought the dvd when SS4 was only 2 when we started noticing him change himself. Little things like when he's with us his favorite color is green (always has been), when he's with his mom he says his favorite color is blue (her favorite color). Same thing with foods, places, tv shoes, superheroes. We figured out that basically anything we had an opinion on she told SS she liked the opposite. Her latest was telling him that if he is bad at her house then he can't come back which of course broke his little heart because he loves seeing her on the weekends. Ever since I noticed him changing I would ask him "How do you feel about that" or "what do you think about that". When I think differently than him (he likes spongebob and watches it at BMs) I tell him that no I don't like spongebob but it's okay, we can like different things. I keep reminding him that he can have his own thoughts, feelings, ideas, preferences. It's all I got. We watched the Dvd once but he's too young to understand what's going one. Your skid is older so it might be more effective.

This happened to me as well, right around the time SO moved in with me a few years ago. YSD did it for a while but no longer ignores me. Once in a while she tries (usually when BM has a pickle up her arse), but she found out it can work both ways and YSD doesn't like to be ignored. We found out that BM was encouraging this behaviour. She too has the "pep talk" with the skids.

EDIT:
I didn't mean to imply to ignore the child like she ignores you.
This is something I do recently, if YSD tries to ignore me. She is almost 13 now! Back then, I would just keep talking to her even if it was only a one-way conversation. I never questioned her why she ignored me (I knew). Eventually she came around and stopped ignoring me.

My ex used to tell our daughters that they were not to talk to us and I have full custody! Yep, Ding Dong really expected 2 young children to totally ignore not only me and my new husband but my entire side of the family.....all the time! Man that man is dumb. Anyway, he pretty much let it slide except when the girls were with him EOW. If they saw my mom out at a restaurant or gas station they had to ignore them and look away. What we ended up doing was coming up with a "secret" signal that they could use so both sides were saying hi and I love you without Ding Dong knowing they were doing it. They even turned it into a game. We also told them that their dad could not controll what we did at our house and they could talk to whoever they wanted too. Yes, we had to remind them of that numerous times but it did work.