Justone..... hang on for a sec......
It's hard being a step-parent and I think you will get a much more understanding response on the step-parent section of this forum: TEENS. IMHO teens can be irritating and appear to be unreasonable, greedy and selfish. Without the natural parental love which is so intrinsic to a natural parent, it's hard to understand how someone who doesn't have that love will react. But it's guaranteed to be less sympathetic.
Think about the times your daughter wanted call in sick from school. I'm sure you were sympathetic and lenient, whereas your DH probably wasn't. This is normal. You can extrapolate that in many ways.
For example, my sister treats her son like little lord Fauntleroy. She takes him food while he plays Fortnite and will drive anywhere to get him McDonalds. She'll order anything he wants on Uber Eats. I just came back from a family holiday to her home country and I was like "omg she is ridiculous"! Then I came home and did pretty much did the same for my DD. And I realised I was doing the same. I'm a sucker for my DD and will do pretty much anything for her.
Being a step parent is so different to being a parent. The level of forgiveness is just not the same, and it's so easy for divisions to grow. Don't be angry at your husband. He's not naturally programmed to love and forgive her when he perceives these slights. It's stressful as fuck, for sure, but do some reading about it....and ask your question again on a forum with people with more experience.
Can I also say - on the other hand, my DH has grandchildren who come around all the time (toddlers). I don't have a lot of patience with them and I can see that my behaviour with them is different to his. He has that natural love for them that I don't have. I do have to grit my teeth at times. Frankly, I sometimes can't wait for them to leave. I can't imagine what it would be like if they were teenagers and I perceived that they were rude to me.
I think the key is to explain to your DH that your DD will gain a stronger self-esteem from a more positive and supportive relationship with him. He needs to know what an impact he has on her young development. This is easily solved, it's just a matter of communication.
Don't think you have to LTB - because, of course, nobody wants that, not least your daughter.
You're a family that just needs a bit more communication. I wish Mumsnet wouldn't be so cruel and full of condemnation sometimes. Some people who are commenting, know nothing about the nuances of living in a blended family.
Be kind and be open. Go play mini golf. Do something as a family.