OP, I have been there - with an overweight, ND child.
1 Accept that the ND IS a factor. Children with sensory processing disorders (and I think the overlap with SPD and ASD is very high) tend to have weak core strength, gain weight on their middle. There is also the sensory act of eating and consuming which can be challenging to manage.
2 if she likes fruit, let her eat fruit. The hand wringing about fresh fruit on MN is ridiculous. Yes, of course, ideally she would not eat an entire punnet of grapes. But it is definitely a better option than many others.
3 Accept that many of the changes can only happen when she is actively working with you to achieve them so your intervention may be limited, In our case, DS didi actually get realyl active in improving things and that helped to accelerate improvements.
Here is what we did:
The first thing I looked at was main meals. I realised that quite often we fall into the habit of "children's food" which are often calorie and nutrient dense foods on the assumption that children don't eat that much. Spaghetti bolognaise, fish pie, shepherds pie, sausages with mash and vegetables etc. BUT, DS was eating large portions of these foods and could happily eat more. So we switched things up. For us, this was fairly easy as we were often eating different foods to the kids (DD is fussy, has intolerances so we'd got into that habit) and DS is not fussy. So meals now can still be large plates of food, but there's a much higher proportion of lean proteins and vegetable/salad. So, for example, grilled salmon with a large portion of stir fried veg and some rice. Or a ciabatta with steak and a large side salad. Or even if doing pasta it's aubergine and tomato, cut in chunky pieces with a bit of chorizo so that the sauce is dominant, not the pasta. Baked chicken pieces with sauteed greens and potatoes. You get the idea. This was transformational and now, years later, we can be a lot more casual about it but we created a process whereby eating large volumes of vegetables or salad to fill up was normal.
Then we looked at snacks. And yes, we'd got into a bit of a habit with less good snacks. This is where DS really had to take part in the process of improving things. We took a sort of at-home Slimming World approach - so things that were clearly treats, we turned into "points" and he could have xx points per day. So he was welcome to eat fresh fruit and veg etc (which on Slimming world would be syns) but his points worked out to just one small packet of crisps a day or a small hot chocolate or whatever. this was important because we didn't want him to feel like he was a) not in control and b) life ws depressing! 
Then exercise: I am firmly of the opinion that a lot of children don't get enough movement and that this is worse for ND children wiith sensory processing issues. I don't have the science to back this up but, for example, at the time, DS was taking part in sports groups. BUT.... in those groups, ND children are often the one lurking on the outside. Doing 100% effort.. but only for 10% of the time. Those groups are often too short in length - 30-45 minutes. So we did a few things:
1 Went swimming a lot more. Low impact movement over an hour or more seemed like a good idea. We also started getting him into the idea of doign a swimathon but sadly that was knocked out by Covid but he was training to build up his endurance and ability to swim multiple lengths before Covid happened.
2 Sought out more active sports groups (this was one he had to be 100% behind though, obviously). In his case, he started playing basketball and the sessions were 90 minutes with almost constant movement. This was completely different to the more "kiddy" sports groups he'd done before (and yes, AFC would, in my opinion, be a great option here for your DD)
3 Made an effort to do more movement at home - this started by him and Dh doing obstacle courses in the garden/house during covid. The key was not so much the difficulty, but keeping him interested so that he was moving constantly, at any level, for longer than 10 minutes at a time - so vs taking him for a walk, which we'd also do, but he'd find a lot of reasons to stop/slow down etc. [Side bar - with DD who also has weight issues but not as severe and it's a bit different, we have done things like trying to reach new heighs in number of sit jumps on trampoline or similar].
As he got older and we moved away from Covid, I was the parent who was allowing my child much more independence than others. But a big part of it was that I saw it as a way to help him stay active, and WANT to be active. So taking a walk to the shop or the park, playing football with the neighbours in the park were things he WANTED to do, and meant he was MOVING a lot more.
DS is now slimmer than most of his friends and is very very fit. He eats like a horse, but we've broken the cycle of poor snacks and the wrong RATIOs of food and he is so active he probably could eat more junk and be fine.
Incidentally, I worry about DD's weight too. She's not as bad as he was and she has intolerances and is a bit fussy, plus we've got some minor medical things happening, but I try, as much as I can, to do similar to the above and we are slowly seeing improvements. Her doctor is also of the opinion that as she grows taller it will be less of an issue. So I hope he's right.