Op Ive spent years negotiating bursaries for my DC;s. IME there are three types of school: those who only offer means bursaries to scholars/high achievers in their entrance exams this is usually pretty clearly stated on their websites, the number of bursaries may vary some will only over a few other lots, the size of the bursary is often dependent on the numbers being offered, e.g. schools which offer up to 100% may only off 5-6 a year means tested bursaries attached to scholarships. The second group make very grandiose claims on their websites about their bursaries, they talk about widening access, offering children whose parents can't afford the fees a chance to have this once in a life time education etc etc. They might even mention bursaries up to 100% but in reality bursaries of this size will only be made in very exceptional circumstances, death of parents etc in general these schools will knock off about 30% max. Thirdly there are a few schools out there who are genuinely trying to move or indeed have moved to a needs blind admissions policy, they are offering very generous means tested bursaries of up to 100% but they are generally very big names (usually very academic and over subscribed) and with matching wealth in assets etc and lots of very wealthy parents and old boys/girls who are donating money to the bursary fund.
If I was you I would contact the bursar again ask to speak to him personally say that you like the school and it would be a perfect fit for you bright DD etc etc then say but I only earn X I dont/do have any significant assets (if you have these e.g. second home, then your chances are not great TBH) and therefore would need a very substantial bursary to send her 75% or whatever (be specific). Say I'm not asking you to commit yourself to me as an individual but before I bring my DD to your open day and raise her hopes I just want to know if this would even be a goer? Make sure he is aware of what year group your DD would be going into bursars don't always listen.to the micro detail. Some bursars offices seem to have this idea that if you attend an open day and really love the school and then find you can't get a bursary you'll somehow manage to magic up the money. This is why you need to speak to them about the details, if they don't want to discuss it I would I suspect your wasting your time.
We move our DC's because of poor academic performance at their so called "outstanding" school, the younger 1 has done very well at his school the older one less well probably no better than in the state sector. Private doesn't mean its going to be more pushy especially at the less academic end of private schools. Also don't be embarrassed if you want a bursary youre going to have to happy talking about your financial situation and what you can actually afford.
It also never ceases to amaze me at the number pf parents who say their school is a 'top 10' or even 'top 5', but if you look into it carefully they're not actually that good, the "top 10" independent schools are pretty unchanged year on year and many no longer in published league tables. Do look very carefully at their exam results, and if possible leavers destinations, anywhere selective should be better than a non selective state school and don't forget some will ask lower achievers to leave after GCSE's and some bus in high achieving foreign children to boost A levels results.
Behaviour isn't guaranteed to be better, not are pupils guaranteed to be more motivated or hard working.
Good luck.