OK, then perhaps some form of childcare might provide what you all need, giving your daughter the opportunity to just play, with no academic focus, and you and your partner could carry on home educating when she's not in childcare.
Forgive me for speculating on your financial situation - reading between the lines I am guessing you are not totally awash in money (you asked about state schools rather than private) but are managing (you are making ends meet on one salary). I don't know whether you are eligible for benefits here; if so, that might reduce the cost quite considerably because you can apply the childcare element of Tax Credits to childcare costs with a qualifying childcare provider.
I used to work part-time while sending my older daughter to a childminder. We used home educating CMs so she would have big kids to play with in the day. But you could use a non-HE childminder and either accept that your dd will just have the company of the CM and toddlers during the day (some big kids enjoy the company of younger ones), or else use a CM in the afternoons while she is looking after other older kids after school, say 3:30-6. There are also "after-school clubs", childcare based usually at schools covering similar hours, with art, craft, outdoor play and games. Though based at schools they are generally run by external providers and would accept children not attending the school.
During the many week-long breaks in the school year there's often the chance to try out sports, drama, dance etc intensively for a week at a time. Alternatively, I don't know how you feel about your dd being subjected to religious evangelism, but some churches run free or nearly-free playschemes in the school holidays with the aim of sharing their beliefs with kids. My daughter was always quite curious about religion and so I was glad for her to be able to learn a bit about Christianity with no effort on my part, plus they did singing, drama, games and cookery.
There are a few home ed centres like the Otherwise Club, which I think is in North London, and where I think you may be able to drop your daughter off for sessions.
If paid childcare isn't affordable at the moment then perhaps your partner might get a job to bring in enough extra money to cover the cost of childcare. Working isn't a total break, but it's different from being with your child 24/7. Or if you find some other HE families nearby then perhaps swapping childcare could be a good arrangement. I used to have my friends' three children one day a week and they took my girls one day, for example. It suited the kids very well because they loved playing together and going to a different house with different toys/food/adults on offer. I was nervous at first about having five kids in my tiny house, thinking it would be chaos, but they were remarkably calm, probably because they were used to each other. The visitors kept my kids occupied far better than I expected: even the days when I was hosting represented something of a break for me. Admittedly I much preferred the child-free days!
This probably reads as if I spent a large part of my older daughter's childhood trying to escape from her!! Most of my home educating friends seem less focused on getting quite so much time away from their kids. As someone who prefers her own company much of the time, I admit that I found it hard at times to spend so much time with an only child who was quite intense and definitely did not want to play by herself. It's all far easier now she is older and does occupy herself with a book, computer or art materials for hours at a time, goes off shopping or to friends' houses or the library on the bus by herself, and likes to go to numerous after-school activities. Her little sister, seven years younger, has always been an easy-going contented child who is happy to potter around playing with toys without constant input from me.
Hope that gives some food for thought.