Germany would seem a bit cheaper as, unlike France, there is no toll (for cars) on the main motorway network. As pointed out by @user3470, you could use ViaMichelin to figure out how much a specific route would cost you. Any reasonable route also seems to go through Switzerland and you would have to pay for the motorway there as well but this would not depend on the route as it is a yearly vignette.
For the rest, I think fuel costs would be roughly equal. However, if you do let yourself drive faster in Germany where it is allowed (you don't have to, obviously), it might reduce mileage/increase costs noticeably. If you have a lot of time, you could also stay away from the motorways and avoid tolls. It's also often nicer landscape-wise and possibly cheaper in terms of fuel cost as well but I don't think this would be practical on such a long journey, especially with kids. Comparing food between countries is difficult but if you are on a budget, you should definitely take your lunch with you anyway.
On the other hand, the most direct route through France would also go through Luxembourg, where petrol is still cheaper than in all neighboring countries (albeit perhaps not as much as some years ago). Now, all this is valid for unleaded fuel, I never drove an LPG car. I did notice that it is available (certainly in Luxembourg, generally also in France, Belgium, the Netherlands…) but never paid much attention if it was available everywhere, how much it cost, etc. A quick web search suggests LPG might be 25% cheaper in Luxembourg compared to France.