Okay, well, I'm not a therapist or anything, but I have 4 kids (ages 4-12), PTSD, and AS...so, let's see if I can throw out a couple of thoughts that might help. Take what resonates and ditch the rest.
One of my kids is currently 7, and he has several aspie traits. I can say that it is difficult to really connect on a deep level with a 7yo, but what your SO is doing to take her with him to places is a huge plus. The key at that age is to clue in on what the kid is most interested in, and try to participate in that some way.
All the kid noise and activity and all that really grates on my nerves, too, so my DH is awesome about handling the routine stuff, which gives me more energy for really focusing on the kids when I can manage it well enough. (That doesn't mean I ignore them most of the time, but DH does make sure I get plenty of alone time so that the time with the kids is more enjoyable for all of us.) Admittedly, that's not as often right now as I'd like but we're working on it.
So, usually when there's a problem with a kid that young (barring any severe disabilities), the problem has more to do with the parents' approach to parenting that particular child (because each child needs a unique parenting approach) than something inherent to the kid. Not that you're doing anything wrong, per se, but that there's plenty of space for you to grow and improve...which is a good thing, because that gives you options.
Consider your family of origin background...what kinds of problems did you inherit from your own family? What kinds of patterns might be subconsciously repeating themselves in your behavior and attitudes now? Take ownership of those issues and tackle them. My family was (and is) severely enmeshed and codependent, so I'm reading lots of books on boundaries (Boundaries and Boundaries with Kids are both great!). A big problem for me is learning how to value my growth process, which makes it difficult for me to accept my children's mistakes during their own growth journeys. So I read books and am getting therapy to help me take ownership of my own issues while learning to accept myself as a very imperfect person. As I grow and become healthier, my kids benefit greatly!
So what areas might you or your SO need to pursue personal growth? Get books from the library, research online...it doesn't have to cost anything to address these issues.
PTSD creates a very on-edge frame of mind most of the time. That can be difficult when living in the same house with a young child (or four). Just like you with your AS, he also needs his own space and time...a place in the house that is a kid-free zone where he can go when he's feeling overwhelmed. It's also helpful for your daughter to have some independent activities--things she can do on her own to entertain herself when the adults in the house are overwhelmed, as well as activities that are specifically hers, like a gymnastics class or dance class...somewhere she can be with other kids and explore her passions. Even at her young age, these kinds of activities give her something outside the home to focus on.
Now, all of that said, she might have her own issues that interfere with having an easy relationship with her. Just remember that at her age, she's not purposely being rebellious, not like a teenager who is ten years older. She is very immature emotionally and, to the best of her understanding and ability, is doing what she feels like she needs to do in order to protect her inner core--not that you or your SO is mistreating her, but that she perceives that her self is not being protected. For my kids, we don't typically punish them in ways that are visible to others in the household. We give them more privacy, which helps them maintain their dignity better. We avoid any kind of shame-based punishment as much as possible. So maybe your daughter needs a different form of punishment? For my kids, sending them to their rooms--not because they're in trouble but to give everyone space to calm down--works better than a more public time-out. That gives them private space to sift through their feelings without offending anyone else in the household. They can come out whenever they've calmed down and ready to act civilly again, so that gives them some control over the situation--even the 4 yo gets to choose when he feels he's ready to behave nicely again (and if he comes out and starts acting up again, we just send him back to his room). Then we come back together and talk about the issue together. "Do you know why you got in trouble?" "How can you handle it better next time?"
I think most kids around this age (6-8 years old) go through an emotional roller coaster experience of sorts. Their emotional selves are becoming more aware of their environment and their lack of control in certain situations. My 7 yo is very easily frustrated. My older kids went through this same stage at this age. A lot of it is just a matter of explaining over and over, every time, how letting your emotions get the best of you doesn't solve the problem. They're not going to hear you the first time you explain this fact of life, or the 2nd or 20th times. But eventually it does start to sink in, a little at a time. Just keep explaining it. Like a postpartum woman, or a teenage boy, or a person going through a mid-life crisis...there are things happening beneath the surface, hormones and other physiological changes, that we just really don't have control over. That doesn't mean we shouldn't be held accountable for our actions, but it helps to have people around us who are patient and understanding while still being firm.
Be careful to make sure she knows that she is not responsible for your or your SO's emotional struggles. That's what starts to cause damage, when she internalizes the disruptions around her that she has no control over. Help her understand the difference between what parts she's responsible for, and what parts you're responsible for. That will help so, soooo much.
