The chance of being offered a place at BHS is vvvv slim due to distance etc but her father will continue to reside in south bucks... can you see what i'm suggesting??...
I sympathise with your problem, and if you're suggesting your DS moves in with her dad, it may be worth considering (if they get on well and he's willing to have her etc etc), but if you're suggesting that you list his home as her main residence when it isn't really, I have to say don't
From your point of view, there are lots of checks and if you're found out, your daughter would risk losing her place, possibly too late in the year to get a place at an adequate alternative. From a wider point of view, it's not fair on those who play by the rules if others get places through dishonest means. However, I realise that may not be what you were asking or suggesting.