Hmm, I had already gone into a kind of emotional shutdown by the time I was 10 because of too much stuff, so I can't go by that experience, but it does sort of remind me of an old friend/ex lover's kid who was maybe in a somewhat similar situation.
She must have been between 7 and 10 back then (she only recently was diagnosed as autistic). Her mother'd always be very concerned about the emotional side of things but in a very direct/pushy way. Like 'I think there's something wrong so NOW we are going to sit down and we won't do anything else until you told me all about it', which in itself would cause her to shutdown, but issues'd also be exaggerated (that's not the really the right word. It was more a situation where she'd be weary of having to go one counselor after the other.) Apart from that, as it later turned out, she'd also be quite concerned about upsetting her mother because she... well, she, the daughter, was herself concerned about her mother, who'd often think she wasn't a good parent (which good parent doesn't?) so she didn't want to add to those worries.
When she was at her dad's, who was more concerned about grades at school, she'd be relieved not to have to stress about that constant threat of being questioned, so she'd be more than happy to say things were just ok, because she knew a simple 'no problem' would do it and she could sort of relax inwardly. So in a way, as far as her dad could tell, there wasn't a problem.
Now what would happen at times is that when I'd be alone with her helping with homework or playing Lego, she would start asking questions or talk about the issues herself, asking how it was for me or if I could somehow relay them to her mom. She really didn't like the elaborate kids parties her parents would organize, and it turned out she didn't really have friends at school; things like that. In a way she really needed time to process the questions and know that answering them wouldn't lead to many more questions and panicking. I suppose I fit the role of much older brother in a way, or it was a kind of aspie-rapport; I'd often talk about my own issues and so with her mother so she knew I needed a lot of downtime myself and that I used to be bullied and the regular stuff. Later on she'd also start to speak more freely during conversations with the three of us, which wouldn't be about her directly, but where she'd be able to again ask questions and that way tell more about how things were for her. But just conversations during downtime making jigsaw puzzles or so, not a forced 'there's something wrong' Q&A kind of thing.
I don't know if any of that can help. I'd say: give her time and space to process the questions, ask again at a second session, but don't push for an immediate answer. She might talk more freely if she's alone with the therapist, because there's fewer people, or because she's afraid of the consequences, or of upsetting you. Setting's important too, especially if she's autistic. Even as an adult I find it hard to find words and concentrate when I'm at the doctor's office (that's why it's important to ask twice, or trice over the course of several sessions). And then there's rapport; it's always easier to talk about things when you know the person you're talking too won't think of it as weird or be accusing, or even better, has gone through the same things. Talk about how things were for you first maybe. And don't panic, overreact or argue; they do pick up on that. Try to stay positive.
Anyway, I'm not a parent so it's easy talking, and back when I was 10 there'd be severe parentification issues and neglect, so roles were more or less reversed and I was the one worried about my parent's issues. I'd always be 'ok' to avoid extra concern or attention.