My parents have a property down near Montpellier and we drive down there at least 3 times a year.
We have done both - driven there and back in one long day AND driven there and back stopping over, or a combination of both.
We have been doing it since our youngest was 18 months, so I can assure you, it is do-able.
However, there are a number of things to factor in.
The French motorways can get extremely busy in July and August. We have seen tailbacks for 10s of miles, adding hours onto an already long, arduous journey. Queues at the peage booths can also be hideously long. Similarly, stopping at services for loo breaks, petrol and/or food & drink can take forever, simply because of the numbers of other people heading south. When exactly are you travelling as you can check the predicted road conditions in terms of expected business in advance. This may well sway you, as if the day is down as a red or black day, I would pretty much say, forget doing it in a day.
The further south you get, the more tired the children become and even though you may only plan to stop for a 10 minute leg stretch, those 10 minute stops rapidly become 15 mins, then 20 and so on.
Also, when you do finally arrive at your destination, it will be very late - everyone will be tired and hungry. Assuming you are staying in a villa, caravan or similar, you would then need to unpack, make dinner etc. All shops would be shut, so you'd have to factor in taking food with you but remember, it will have been in the car since the day before with the ferry crossing and journey down....
From our experience, the best time of year to drive down in one day is at Easter time.
Ask if there is anything else I can help with - I don't mean to be the voice of doom and gloom but you want to enjoy your holiday and if you gear yourself up for a MEGA stressful drive down, it may take you a few days to get over it and spoil the first bit of your hols.
Hope this helps!