I have also worked part-time in a task orientated job and been asked to swap my days around.
I used to offer the option of paying for additional childcare or joining the meeting via a conference call (this was on the proviso that my childcare was cbeebies!). Any serious hours I always took in lieu but the odd hour I never bothered about.
If they wanted me to come in, I used to get an invoice from my childminder for that particular day and submit it as expenses. (OK I used to write the invoice for my cm and just get her to sign it, so that she wouldn't have to do the paperwork.)
When I submitted my original application I did include a paragraph that I would try to be flexible and I needed enough notice and additional childcare costs paid for (because I usually swapped days rather than work additional days, so still had to pay for childcare on my in-lieu day).
So it is not unknown, but we had the technology that meant I could do exactly the same work from home as the office except a face-to-face meeting.
I will also say that as time went by and everyone got used to my hours, the meetings were arranged around my "in the office" hours. For example I could never do a 2:30 appt on Thursday or Friday but I could do earlier in the office or after 4:00 by conference call.
Can you come up with a couple of alternatives, either on your working day, conference call, they pay for childcare or during the school day if your children are old enough. If you can come up with some different options then you should be percieved as trying to work with them for a solution.
P.S. I should say, that they were VERY keen to keep me at that point, just long enough to pass on all my knowledge to the staff in India as part of the off-shore contract, before the last round of redundancies. So I was working from a position of strength which I shamelessly abused in getting exactly the hours I wanted in the first place.