I would suggest go to counseling. There is notich you can do if anything especially if the parents aren't addressing her behavior.

She also needs to deal with her parents divorce and the fact that her mother chooses to be in a same sex relationship and despite her feelings about that she doesn't get to dictate how her mother lives her life.

The most telling sentence in your account is "my girlfriend is completely controlled by the daughter".  Until this changes, and your GF asserts herself as the proper authority over this child, there will be nothing you can do to change how the girl behaves towards you both.  I would suggest that your GF seeks professional support to do this, if she cannot manage it on her own. Or just watches some episodes of "Supernanny" or similar.  The girl should not be allowed to come between you, you need to present a united front and give her less power.  Children don't want power, it unsettles them, they need firm boundaries. Your partner needs to understand this. Maybe she herself is unsure in her choices and the changes she has made in her life, this is reflected in her daughter kicking off. 

The daughter isn't damaging the relationship, your partner is.

It's HER responsibility to parent and discipline her daughter. It's HER responsibility to tell her daughter that she has to be respectful. It's HER responsibility to get her daughter therapy if she is struggling (and it wouldn't be a far fetch for a preteen to struggle with their parents' divorce, new relationships, AND affirmation of sexual orientation). 

Whatever guilt your SO and her ex feel is preventing them from being good parents, and in these kinds of situations, not being a good parent inevitably leads to them not being a good partner.

You didn't break this, so you can't fix it. My suggestion is two-fold. First is to disengage from the daughter entirely. Second is to tell your SD to stop behaving disrespectfully to you:

"SD, stop fllipping me off. It's rude."

"SD, move. You're not sitting between your mother and I. It's rude."

"SD, stop talking. Your opinion isn't warranted here."

Then to your SO, you say:

"Her behavior is unacceptable, and I'm disappointed in you for not sticking up for me. That hurts my feelings, and it hurts SD in the long run by not being parented. While I am appalled by her behavior, I put the blame on you."

This should be mixed with a heavy dose of you walking away when SD behaves poorly and leaving your SO to deal with her daughter "winning".

If you're saying to yourself "but I don't want to do this and it hurts my relationship" or "I'm non-confrontational", stop it. The truth is, SD's behavior is a symptom of a bigger parenting problem that is already causing problems in your relationship and SHOULD lead to you walking away from your partner permanently. The only chance you have of making this relationship work is pointing out all the crap you put up with and laying it at the feet of the person who can fix it - your SO. If they don't fix it and only get annoyed, threaten to break up, the relationship degrades, etc then all you've done is sped up the dying process on an already terminal relationship. That is a GOOD thing while you're dating so that you don't end up entangled in a marriage, joint assets, and a possible child. 

Your role in this is not to make SD like you. Your role is to hold your SO accountable for her parenting failures that impact you. If she won't take responsibility and begin fixing these issues, then your SO ruined your relationship, not SD.

Your GF is the problem here, along with the child's father. They've raised a kid who thinks she can do whatever she wants and is on level with the adults. This will not get better without a change in parenting, it will only get worse as this girl ages.

Your GF would have the same issue if she was dating a man, the same-sex part just give the kid a convenient complaint.

I'd rethink this relationship. You may think your GF is wonderful now, but you won't after a couple of years of this nonsense.

"healthy relationship"?

I don't call a partner who does not discipline their child and who allows them to flip off an adult, be rude, cause arguments and is ignorant as a healthy, loving partner. If your OH and her ex don't get a handle on sd11 anger and if they fail to do what they are meant to do - actually parent sd - then sd will be out of control as she flexes her power and realises what she is permitted to get away with. Her outbursts and behaviour will only escalate as she ages and your frustrations and resentment at being so badly treated will also balloon.

Either your gf starts showing you she can parent or I would leave as relationships like yours are doomed or at the least in 5/10 years you will bitterly regret remaining and wasting your time.

My lesbian friends would love this post, because it just so illustrates one of the stereotypes of lesbian relationships: getting in too deep too fast. I really feel for you girl, because this is a circus.

Blending has the highest chance of success when it's approached in a slow, considered, methodical manner. Timing is EVERYTHING, and you've entered the scene at a terrible time. I'd go so far as to say your gf has no business dating ANYONE currently, because she's got so many problems she needs to sort and her kid needs her full attention right now. This tween is struggling because she's been expected to handle a lot of serious stuff in a short period of time: her parents' divorce, dad getting engaged, the new norm, AND her mom's sexuality. Then she's presented with her mom's first same sex partner, a sort of cherry on the sh!t sundae her life has been of late, so you make an easy target for all of her angst.  I'm not excusing any of her behavior, which is exacerbated by cr@ppy and diluted parenting; just trying to offer some insight on her p.o.v.

As things are, there's no room for a healthy relationship because your gf didn't do the necessary work to prepare a place in her life for someone new FIRST. But that's on her, not you. You can't fix this mess, because you didn't make it. 

As others have pointed out, shi!!y parenting kills love, every time. You HAVE to take care of you, so have a serious talk with your gf and give her some ultimatums. If she doesn't seriously up her parenting game, get her kid into therapy (individual & family), and start protecting you from HER baggage, then walk away from that dumpster fire. There's a smorgasbord of other women out there who have their act together, many of whom have no children, so set limits on how long you're willing to wait upon improvement.