It is certainly possible. Some people tour with single speeds. However it might be hard going depending on your route choices, amount of weight you're carrying and lowest gear you achieve.
Personally I prefer to have a lowest gear of around 22 gear inches which translates to a 34 tooth front ring and 42 tooth largest sprocket at the cassette. With a single chainring at the front you won't likely be able to achieve that since the front chainring would have to be tiny.

Also you did mention you did not like changing things. To me that's a non productive stance, since you'll have to change parts when they wear down anyways. The whole drive train is made of expendable components which will need to be swapped at some point due to wear. Also you don't need to toss a components when you replace them with something more suitable for touring. If you add a triple crankset for touring use you can put the original crankset back when you've returned from your tour.

Just a word of caution. While the Dawes Fox is as suitable for touring as any bike and you will likely be able to cross France with it no problem, keep listening to what your body is telling you. Less gears and higher gear ratios will mean more strain on the uphills. This means you'll need to grind instead of spinning which can cause wear on your joints, especially knees. Now I am NOT saying you will blow out your knees by riding a bike with few gears. I AM saying that you need to be alert on pains and aches and just push the bike uphill if you feel like going at it with tenacity is going to do more harm than good. There's no shame in that especially if you don't have enough gears to get really low gear ratios. Since touring is not a competition there's no rush and pushing your bike now and then lets you stretch your legs and freshen up a bit.
If you develop knee or back pains that worry you, take a rest day or two. Also remember to have regular rest days to recover and eat enough.

And one last thing. France is a beautiful wine country with a selection of some of the finest wines on the planet. Just try not to partake too hard into the wine tasting since alcohol will be quite detrimental to your sleep quality and recovery even in small doses. Learned that the hard way myself