RobinMoiraWhite
I think that you may have had a hand in creating the document below, in which case you will be well aware that many of your colleagues differ in their interpretation of the EA2010, both in terms of its definitions and, particularly, the correct comparator in cases of discrimination of those with the protected characteristics of gender reassignment.
committees.parliament.uk/writtenevidence/17849/html/
Given that paper, I'm sure you don't mean to imply that this is is settled law (at all, but certainly not in the way you might wish).
The implications of that uncertainty for the scenario in the OP are quite striking in that there is most likely no course of action open to the school that could give them an acceptable degree of comfort that they were acting on the right side of the legislation. At some point I expect there to be a high profile case resulting in clear, binding precedent, (most likely, regrettably, based on a serious incident of some kind).
Given it may well be sued either way, if the school's policy decision lay with me, I would be inclined to step back from a black letter legal debate at that point and take such of some wider considerations, including:
a) the weight of the increased risk of (i) sexual activity / assault / rape / pregnancy to which the school was exposing young girls in its care as against (ii) emotional upset of a transgirl being refused access to the sleeping quarters of those girls; and
b) the relative impacts on reputation of the school should one of the 12/13 year old girls in fact engaging in sexual activity or being assaulted/raped/impregnated by a transgirl (assuming - generously - that to be the pupil in question's genuinely felt identity) following the school's decision to allow a transgirl into the single sex sleeping area (and without informing parents) vs being sued by a trans pupil for discrimination in refusing their accommodation request.
Test cases remain to be heard, of course, but I know which position I, as a school, would rather be defending from!