Will a Dutch oven be a good investment in a motor home with a butane gas burner?
I don't think that it is a good investment. Without a real oven, you won't be able to bake or roast in the Dutch oven. Unlike historic applications, where the Dutch oven was buried in embers and could warm the food inside from all directions, a Dutch oven on a gas burner functions pretty much like a thick-walled pot. 

Using it will let you make stews, soups, and some sauces. If you want to make sweet things, you will able to make cremes and syrups in it, but without an oven, you are limited to only combining these with fried dough and griddle/flatpan items like crepes. Many classic types of pastry require baking in an oven. And for both sweet and savory, there is not that much difference between a Dutch oven and a normal pot. It will give you a more even heating, but it will respond sluggishly to temperature changes in the burner. An advanced cook can use this property well, but if you are limited to a very small selection of cookware, there is no need to get both a pot and a Dutch oven, and out of the two, the pot is cheaper and easier to handle. So, my recommendation: get a pot instead. 

If you want to cook more and your budget allows it, consider buying a toaster oven with a 30x30cm cavity. These take up very little space, and are cheap (mine was 50 Eur). They are good for baking practically everything - pizza, cakes, roasts, etc. I have never had a regular oven since I left my parents' home 11 years ago, and have made some pretty advanced stuff in the toaster oven. It is definitely a good investment.