I think it's a great idea

I think that in approaching the school, you need to try to predict ahead of time where this particular school's potential objections might be. A few I can think of:
1) Scheduling - it will help if the school has a standard schedule they follow (either every day or every week), so that she's consistently missing the same subjects.
2) Income - will this impact the school's income in any way? For instance, if the school is only partially full (has openings in her grade in particular), and if you're only giving the school the option of either letting you do this or you pull your dd out completely, the better financial option for the school is to let her enroll part time. OTOH, if the school is full and has a waiting list, then the school might object to you pulling her out unless you continue to pay tuition - because they have other families waiting in line to fill those slots. One student's tuition may not seem like a lot to a parent, but in both of the private schools I've been associated with, each students tuition actually does matter.
3) The school is concerned that if you do this, other parents will want to also. I don't know the best way to counter this, other than a sincere doubt that other parents would - it's not a typical situation and it's also not typical for a parent to have the flexibility and time to be able to do this.
4) The school thinks that pulling your dd out for part of the day will make it difficult for her to be socially integrated with the rest of her class. I think this is a weak reason for not doing it, since she'll be in class with the rest of her class for the majority of the week, but mention it mostly because it was brought up as a potential issue by school personnel when a friend of mine was partially homeschooling.
Good luck! I hope you're able to work this out!
polarbear