We've done this route a fair few times. Oxford to Lauterbrunnen (then on the train up to Wengen) which is just a little bit further up the road from Interlaken.
Our kids are now 11, 9 and 6 but we've done it about 4 times at varying ages. And many times before kids.
We've done it stopping overnight around Strasbourg, and in mid france but our preferred journey is leave straight from school on the Friday and drive to hotel at folkstone, get on the channel tunnel train (using tesco clubcard vouchers) around 6.20am. Then it takes us about 10 hours to drive to Lauterbrunnen so we end up in Wengen in time for a late dinner.
Ours have never minded long car journeys as we have other family that lives near Inverness and with 3 kids, driving to both scotland and switzerland is much much cheaper than flying.
It's motorway all the way to Interlaken so you won't need to worry about driving in the snow, and at Christmas it's very unlikely there will be snow on the ground in Interlaken itself, it is surrounded by mountains but is very low itself. From Interlaken you can get the train up the mountains if you want to get to snow.
We tend to feed them in the car, we drive for 3 hours, stop stretch everyone's legs, have a run around and then get back in the car and eat our sandwiches etc whist moving. I don't see the point in stopping to sit down and eat as you're already sitting in the car for hours and hours. We also both drive so we can share the driving.
Getting a Liber-T tag for the tolls on the french motorways. Means you can use the lanes with a T above them at the toll barriers and can sail through without stopping, it automatically blips your tag attached to your windscreen and you will be billed at the end of the month. The queues can get very long otherwise if you're travelling to/from Calais in the holiday period. You will also need a tax vingette for the Swiss motorways.
We've never had in car dvd players but lots of people do. If you don't mind driving and your kids travel well, then go for it, not mad at all. By the time you've allowed for queueing at airports and boarding etc and then got the train to interlaken, it really isn't that much longer journey time to go by car and you're not limited in amount of luggage etc that you can take, and it's cheaper!