I sympathise with your dilemma now that you've given your dd2 this opportunity and she hopes to go to the private school, but I think you need to sort out in your mind what your dd1 failing that entrance exam actually means. Obviously it means your dd1 couldn't go to that school, but does it have to mean anything more than that? Are you taking it as a sign that spending say a few thousand pounds a year on her education in other ways would be wasted, because she wasn't good enough to pass that one exam?
If you say that because your dd2 passed that exam she gets to have £12k or whatever spent on her education each year, and your dd1 now gets even less money spent on her, then you're making that single private school exam an incredibly significant measure of what they each 'deserve', or need, educationally.
All a private school is is a convenient 'package deal' way of buying in lots of educational resources at once - facilities, teaching time, smaller class sizes etc. It's not the only way to spend that money though. The whole may be greater than the sum of the parts but some of the parts are available separately - through things like extra tutoring, trip opportunities, extra-curricular activities.
If you let your dd2 passing that exam stop you spending money on your dd1 that way, by directing all your money towards the 'package deal' private-school education for your dd2, you are favouring your dd2 hugely and your dd1 will be aware of that.
Private for one and state for the other is fine if you're still directing your resources fairly equally towards each child, however you're doing it, but if you do this you'll be directing all your resources towards your dd2 based on her passing one exam, with virtually nothing left over for your dd1.
It isn't even that clear that your dd1 not passing that exam isn't a sign that she needs extra help more - maybe of the two of them, your dd1 might be the one whose life chances will be most significantly improved by money being spent on something like extra maths tutoring or the chance to go to a language school in the holidays? If so then she's going to be doubly let down by everything going to your dd2.
Like I said I do sympathise, but I think you should think seriously about the weight you're giving to the passing/failing of that single exam, before you let it direct your spending for seven years so extremely towards one daughter and not the other.