I wouldn't agree with a pp that "most schools only offer 8 GCSEs".
Some do. One of the top rated schools in my city offers 8 unless you are able enough to be chosen to do triple science, then you do 9. Either way you get one choice as everyone does MFL (only French offered) and history/geography.
So at that school OP, your dd would not have Citizenship taking up an option (which I would agree with others is not the case here anyway) but she would still have to choose one out of DT and business.
But I know plenty of schools where students doing double take 9, and those doing triple take 10 - my DCs' for example. Tho even there they only have a choice over two GCSEs.
Option GCSEs IME as a teacher typically take up about 5 hours per fortnight. I certainly wouldn't want to teach my subject (MFL) in any les than that. I imagine that citizenship at your DD's school will be an maybe two hours a fortnight. It really isn't taking up a GCSE slot. It's more than they have to study it so why not get a qualification?
Schools vary in their approach - some say, may as well take RE GCSE, some citizenship; some (where I taught) take the KS4 kids off timetable for a day twice a term to cover PSHCE (yes it includes citizenship). My DCs' school covered both in assemblies and tutor time. Some schools local to me have a day that starts at 8.45 and finishes at 4pm; some start at 8.30 and finish at 20 to 3. There is variation.
I would say that you need to find out what the "wiggle room" is, suggest to DD that she does triple science and don't bother with business, which is not needed at GCSE for A level. What is the DT subject she wants to do?
Great posts upthread from Teen and mintycedric btw.
Sorry reading more of the thread, I see your objection is not to the lessons in citizenship, but to having to do the GCSE. So would it be better if she didn't do it? She wouldn't be able to do another GCSE because there won't be any lesson time! So she would have 9 (totally fine) instead of 10; those 9 could be, if I read you aright:
Eng lit
Eng lang
maths
biology
chemistry
physics
Spanish
history
DT
Totally fine! Just ask the school not to enter her for the citizenship GCSE I guess. Though why I really still don't see.