I would say 2 things - firstly, there has to be a deadline. You were given it. It isn't obligatory to give a letter about individual pieces of coursework from individual subjects, just that kids are told they have the right to appeal the marking (not the teaching, just the marking) after they have seen the marked version. If you don't meet the deadline, you miss out....same as opportunities for re marks of exam scripts - there has to be a deadline and they can't budge on it.
Secondly, schools don't tend to mark down. They tend to over mark, because of course they want their kids to get the best marks possible. Where the Board alters coursework marks it tends to be marked down, rarely up. When you ask for it to be re-marked you are asking the school itself to remark it - you would only do this if your child has looked at the markscheme and thinks it has not been applied properly - ie it is not annotated in relation to the markscheme etc.
Be aware marks can go down as well as up when internally remarked.
So being a bit disappointed in the mark or feeling a student wasn't well supported (and it is an Independnet research essay - so whilst support is given, the research and writing up has to be independent) isn't grounds for asking the school to remark. Feeling the school hasn't internally moderated the marking or has not applied the markscheme is.
So, you have probably missed the boat on this. Even if you hadn't, you would need to be pretty confident there was negligence in marking to go for a remark, and unless the school is doing a bad job, won't get more marks. Schools have to offer the opportunity for a remark and it is right they do have to, but the expectation is that few students will take up that offer and it is there for when crap marking occurs.
Lastly, remember that often half of the coursework goes off for external moderation. Another 'check' is built in then - schools are commended or ticked off for their marking and marks are changed by the board - as I say, usually down and rarely up.
I think you've missed the boat and it's not really clear why you want a re mark. Are you saying you think the school was incompetent in its marking and didn't apply the markscheme? That really is the grounds for a remark....but only if you meet the time criteria. Schools have to meet the requirement to tell people of their option for a remark and a timescale, but students have to meet the deadline requirement too.