This is extraordinarily difficult and I completely sympathise. DS was very overweight at a similar age, including some stealing of food and also just the relentless begging and complaining and whining and we gave in way too often (DH was particularly bad and in fact, it caused some tension between us as a result).
It's also worth noting that DS has Sensory Processing Disorder and, as it turns out, inattentive ADHD. weight problems in both of these is not unusual.
DS himself started to realise he was overweight. The other children teased him. So there was incentive on his side to get improve things. Which allowed us to all work together. Unfortunately I think it's very very difficult if you're imposing it on your DD without any interest from her. Having said that, here are a few things we did and that worked for us:
we knew that cutting all the treats etc in one go was going to be too hard. So we created our own sort of version of Slimming World Sync. We said he could have an allowance of "treat" foods equal to 10 points and we allocated those points based on SW (only because we had Sw data so it was easy - you can do this however you like - eg we didn't include healthy fats etc like SW did. We just did this for obvious treats). This then dropped later as he started to see the results. DH also implemented a "you can have as much fat free greek yoghurt as you like" like Slimming world recommends. I wasn't wild about this at first, but actually, him having yoghurt when he felt hungry - sometimes with a weetabix or something - wasn't a bad thing.
Main meals - he always ate his main meals, which was one of our problems - he at anything and everything! But we tried to make main meals very filling, very nutritious but less calorific/fatty. we moved away from "kids" food of oven food or spaghetti bolognaise and fish pie and all those nutrient-dense, high calorie food you give children because they so often eat so little of it. Lots of grilled/baked chicken and fish with huge piles of vegetables and served with potatoes/rice. Luckily, he is a good eater so he liked eating "grown up food" with us at night (a bit later than DD was eating) so this wasn't hard. I think the food was interesting enough in flavour and texture as well that it satisfied him.
You say she's active, but what does this entail? DS was pretty active, but... because he was overweight, he would stop a lot - so he might be outside or whatever, but if you watched him, you'd realise he was taking lots of breaks, not pushing it etc. If we went for a walk with the dog, he would dawdle behind me, not keep moving etc. He wasn't really as fit as you might think. So him and DH got an exercise program going where they would do circuits in the garden (this was during covid) and they would work to constantly improve - so 5 circuits without a break, then 7 etc. Or a harder circuit or whatever. They got hilariously creative with circuits involving running up and down stairs, push ups, jumping on and off the trampoline etc! When gyms etc reopened, he did a lot of swimming - it's a nice low impact sport but kept him physically moving and his weight was less of an issue.
Good luck, I really do know how incredibly difficult this is and I sympathise a lot. A friends is in a similar situation with her DD. They police everything and snacks and treats aren't kept in the house but it's nonetheless relentless - I've seen this child go round tables stealing food from anyone she can, for example. And because she IS so overweight, getting her to do the kind of longer-term exercise that would support weight loss is very difficult.
DS is now 13, tall for his age and very slim. He spent the last couple of years fine weight wise, but possibly at the top end of fine. Then he grew half a foot over six months and is now super slim! 