Really feel for you. It is so expensive around here. I am afraid I cannot comment on the state schools as our two go private. However, from reading posts on here, your 4th choice may well be better than you think. There seems to be a lot of gossip about schools being 'bad' which in fact may not be true, or as bad as you think when you visit and research.
I would think very carefully before launching into private schools. I have no regrets and we are very happy but it is very expensive. If you look at the fees now and feel you would be hard-pressed, I can tell you that when it comes to it, it will be even more. Fees increase every year, and there are untold extras, such as uniform, after-school clubs, school trips, music lessons. I would not forego your one family holiday a year. You may have to down-scale it a little but it's important! Private school is great, for many reasons, and while I think it's certainly worth tightening your belt for, it's NOT worth years and years of watching every penny (and it gets worse every year as fees will increase as they get older). If your 4th choice state school does turn out to be not right for your girls, yes, maybe think about selling up and moving a little further away. You have probably made some money on your house too since you have been there. It's amazing how much more you get for your money by moving only half a mile or so. You may then get a great state school, or have more breathing space to pay to go private. Yes, it does cost a lot to move, but it's a one-off cost, which may make sense in the long run.
I am sure others will have some good ideas too.
Good luck, and don't get too disheartened. Either way, you are sitting on a valuable property, and you have choices.