Definitely worth negotiating, all they can say is no! It seems that they do want your DC and are willing to help towards the cost so you are half way there already. When we were applying for DS, we got offers from two schools and played them off against each other (discreetly!) as they both wanted him! End result is that we got him in to our (and his) favourite at a better bursary rate than they started off offering so we are all happy now.
I'm not sure I'd ask for a breakdown as that seems a bit confrontational - afterall these things are given and are not something that you have as a right.
I do think it is worth explaining that you really want the place at the school though and letting them know that your financial situation is perhaps worse than it appears on paper because of the additional costs you have for DS3. You could offer to supply a breakdown of your outgoings to give a better picture and see if it could help.
I think you need to work out what %age bursary you need to make it sustainably affordable. Then you know what you are negotiating over.
If the bursary pot has been divvied up between all bursary pupils already, then it's possible they will have no more funds to allocate. But as you cannot possibly know this, it's definitely worth asking. They may say no, but if you cannot afford to take the place on current offer, you really have nothing to lose by trying.
As your DS has a scholarship too, and will be expected to bring credit on the school, you are probably in a good position to negotiate. Anyway, you have nothing to lose. Ask for an interview, explain how much DS3 costs, and how much DS1 loses out because of the attention his brother needs. And emphasise how wonderful the school is. Good luck
Don't post on here much as usually on other boards but could do with some advice. DS1 has won an academic scholarship and been offered 55% fee remission to a local private secondary school Private schools are all very new to me and its been a learning curve - its only when we have got near to the end of the process that schools are starting to reveal how much they give in bursaries and even now I don't know how they calculated the figures. A major motivation in applying is that as well as it being a good school for DS (academic, small classes) we have DS3 (a severely disabled child) so good social reasons why a school that can offer extra activities etc will mean DS1 will not miss out so much - and to be frank take a lot of pressure of us as DS3 takes up 95% of our time. However the 55% is no where near making it affordable - I don't know if they have included all of DS3 DLA money and if they have taken into account the very substantial costs of raising a disabled child eg massive travel expenses to therapy etc. So is it acceptable to go back and question the % and ask for a breakdown / check if there is any additional discretion? I don't want to make a fool of myself but at the moment the costs are totally out of reach.