I think that £200 a day is way too much - that sort of fee would really send a message that you didn't want to do this.
I would calculate as follows:
flying time 2 hours
check-in time 1.5 hours
time to get to the airport - whatever it is from where you are, but say, 1.5 hours
time to hang around waiting for return flight - whatever it is but could be as little as an hour if it is a UK airline and you are going back on the same plane that brought you out
return flight 2 hours
time to get home - again, whatever it is, but let's assume 1.5 hours
That makes a total of 8 hours plus.
Charge this out at a nanny rate and a high nanny rate to compensate for the inconvenience and the fact that it is a weekend: 8 hours x £12 = £96, call it £100. Even at £15 an hour, it would come to £120 - not £200.
Obviously, if the trip takes longer than the 8 hours estimated - for example, if planes are delayed, you charge for time spent. You can give her an estimate of the amount but do not cap this. You should not ask anyone to take you to or pick you up at the airport but get a minicab, keep the receipts and charge them. You shoudn't have any other significant expenses as it just a day trip but agree in advance that, if for weather or any other reason, you are forced to stay overnight in France, they will pay for a sensible hotel. (We got diverted for weather reasons one memorable trip to the South of France, which ended up a 24 hour journey...)
I wouldn't worry about business vs. economy class - it really makes very little difference on a 2-hour flight.
If they are not willing to agree to these terms, I would refuse.
In defence of the mum, I must confess that we have done this ourselves (Dad, 6 month old and I) Once you have done the trip, you ask yourself what on earth you were worried about but it can seem daunting to an inexperienced parent!