You need to take a big step back and stop everything that you do for SD that includes buying her clothes, doing her hair, choosing what she wears. She's made it clear that "it's not her style" and both your H and BM are siding with her, so for your own sanity, disengage from this. You will save yourself a lot of stress and frustration. Take care of your child(ren) as you see fit and allow your H & BM to parent their daughter.

He says he can't do it without you but he's not supporting your efforts. He's also not "alone" the BM is in the picture so he needs to put his big boy pants on and do the work.

Once you detach you'll be thanking your future self and wondering why you didn't do it sooner.

If your SD is going behind your back and undermining you, the BM is undermining you, and finally DH is not backing you up, you are what I term "dead in the water", like a sailboat with no wind, trying to head for shore, the tide blowing you further out. Stop fighting these tides. They will only pull you out further, causing more resentment.

Withdraw - they DONT WANT YOUR HELP. This girl has 2 parents and you are not one of them. Im sure its fun, girls can be fun for a mom, but she clearly doesnt ascribe to your vision (right now). SD might come around, seeing as how you are culturally similar, but not right now.

Interesting, how you mentioned the cultural connection. Im Anglo, Dh is Filipino, his ex whom I call Toxic Troll, is also Anglo. We actually have some similarities, as far as hair and skin. DH and SD's identify heavily with their culture. I have learned to make the foods they like and am looking to expand my repertoir in that area. TT has never really connected with that "other side". Not really. And the family gatherings are huge and epic. With all the different flavors as everyone has intermarried with culturally different partners. So I understand that you wish to connect with SD in this way - it IS special. But you need to give her space. Let her pick out her own stuff.

My SD13, Munchkin, when we go shopping, we do resale, I give her a budget, and she picks out her own stuff. Its not really anything I like (she is currently into overly large shirts, and t-shirts,, and ripped jeans with converse, and her lovely thick shiny hair is cropped very short in a bowl cut) but its what SHE likes.

Take yourself out of the financial equation. Dont parent her. Allow DH to step up to the plate and parent her.

I agree with the stopping. 

If her hair looks awful because it isn't properly being taken care of (I am make a big assumption based on the cultural differences you stated), that is on BM now. 

If she doesn't like the clothes you got her and has her own style now, let your DH take her shopping, just make sure that he understands and will keep within the budget you have agreed on. 

As for teaching and "emphasizing your culture too much"- screw that noise. That is your house and you are raising 2 other kids in it. BM doesn't get to make that choice. 

Do not do anything for SD. Do do not drive her anywhere, Do not buy her anything , do not spend your money on her.  It’s all up to DH to take care of her.  You are always wrong So now it’s all up to DH to cook for her clean for her ect.  No going places with her. No vacations with her.  Ect.

I'm guessing that SD has reached that age where she is "aware" of her clothing more.. what other kids have.. etc.  Unfortunately, I'm not sure you remember exactly what it may have been like as a young girl who feels like if she doesn't have the "right" clothing.. that she won't fit in in her community.  While I am not big on rewarding kids for "bad" behavior.. the girl obviously has tried to tell you that she would prefer different styles than you have bought for her.  So.. yes.. she seems ready to be part of that decision process.  Unfortunately, there is no amount of "suck it up buttercup" that she will want to hear that will make her want to wear the clothes you bought for her. 

When she tried to explain why she didn't want to wear what you bought her.. you kind of gave her a "flip" answer.  While it may be true that she can buy what she wants when she has her own money.. there is also something to be said for kids having some say in their clothes (within reason of course).  She complained to your DH and her mom because she doesn't want to wear what you bought her. 

Now.. she should have told you when you brought the stuff home that she didn't like it then...when you had time to return it.  But she didn't ... probably not trying to make waves.. but now when the pedal is to the  metal.. she doesn't want to wear them.. and wants some new clothes instead.

This can be a hill to die on..or you can disengage and tell your DH he and his EX can take care of this kind of stuff from now on.  If he bought her a few new things from his own money?  that's on him.. as long as it doesn't impact your bank account.

SD13 stopped wearing ANYTHING I bought her this past winter, even though it was nothing but the leggings or underwear she usually wears.  I considered it a power play, so no more clothes from Cover.  I stopped with major help with clothing years ago, but occassionally would pick somehting up here and there but I was done with it around February when she refused the leggings. 

Disengage - she's made it clear.

Even as you must disengage, the other benefit is that you can spend MORE time with and on your Bio. Have a very firm stance on the fact that this is YOUR household, here YOU are QUEEN BEE. There can be only one Queen Bee in any hive, and you are IT. BM can go stuff it if she thinks she can controll your household.

Now telling SD she doesnt need to respect you - your DH REALLY needs to step up his parenting game and back you up. That means when SD disrespects you, there are repercussions. Taking away of electronics, etc. You know, punishment for bad behavior kind of things. If he does not back you up in this, then he obviously doesnt need your help. Disengagement helps with this tremendously.

I had much resentment until I found steptalk. Keep posting and reading - it definitely helps to understand the dynamics, and regain your sanity as well as control of your household.

I don't know how old SD is, but maybe she's just at that age where they all start cringing at the clothes we pick out.  It's part of growing up, figuring out who they are, and yes...developing their own style.  That was a hard shift for me too with my DS, but it's more than likely normal behavior and not meant to be disrespectful.  

"But DH took her school shopping anyways. I lost it! I don't feel that is right for kids to override parents decisions." 

It sounds like DH is actually the one who did the override of your decision, not SD.  There's an easy fix to all that.  Like other posters commented...leave the clothes buying/shopping to the parents.  Done deal, on with the day!

As far as the cultural part goes, I'm multiracial and all I can say is it's a very individual thing for a person to decide what that means for THEM.  I do wish my parents would have shared more about my cultural heritage with me, so I think it is a positive thing for you to express your culture as you wish in your home and share that with SD.  However...I definitely bucked the system with the one out of 3 heritages they pushed upon me.  I don't even identify with that heritage at all anymore, which is probably unfortunate.  So, I guess just be careful not to push too much or your good intentions might just backfire and give rise to aversion.

"What to do when you're battling SD, BM, & DH?"

 You step out of the equation completely, because it's not your kid and not your war. The only way to win is not to play at all.

Instead, focus on what IS in your purview: your home, your bios, and your boundaries. Let go of other people's kids, because as you've just been shown you're damned if you do and damned if you don't. 

(((HUGS)))))

Good advise given above.

Want2----maybe I am not understanding what you wrote.

why should OP waste her money on clothing the child refuses to wear. So you want OP to go back and buy more so sd can refuse her gifts again?. Same thing if op cooked dinners and over and over again skid pushed the plate back, refusing to eat. Do you think op should keep cooking for sd?

That is ludicrous.

OP is not a doormat.

 Sounds like she’s at the age where she wants to pick her own clothes. And as long as she’s supervised doing so, I think it’s wise to let her pick her clothes.

And if the parents are disagreeing with how you are raising their child, then let them know that they are on their own and they can raise their child them selves. It’s rare that it ever works out raising a stepdaughter when there’s biological mom in the picture that doesn’t want any kind of help.