I live in a Motorhome and only have a butane gas burner. Cooking over a butane gas burner is all I have and is limiting using just a pan for recipes. Does anyone have experience investing in and using a Dutch oven on this type of little butane gas stove? Any suggestions on recommend cookware is welcomed.
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.
If you have access to electricity, an inexpensive slow cooker from any discount retailer would be more economical than heating a cast-iron pot with butane, and can easily handle the stews and braises typically associated with the dutch oven. I'd save the butane stove for a small skillet... the local restaurant supply store will have inexpensive models of excellent quality... and use that to sauté and sear the ingredients as required before putting them in the slow cooker.
It is possible to bake on the stovetop, if that's what you're looking for. A cast iron dutch oven, which will take a while to heat up and cool down, will likely be pretty helpful in this kind of baking - with the caveat that you would treat it like an actual oven not a pot, and you would be setting your food in a smaller pan or tin, on top of a rack or trivet (very important, so the heat is indirect), inside the preheated dutch oven - so the direct heat doesn't burn the bottom of your food. Ovens are usually directionally heated anyway, with the food suspended in the middle so it cooks by radiated heat and doesn't burn from direct contact.
You might also want to look at indirect heat or residual heat baking methods, where you preheat your oven, or dutch oven in this case, and when it is hot, turn it off before adding the food (small, individual items; something flat and quick-cooking [like pizza]; or a one person serving of something precooked), and let the residual heat warm up your food - I've used the technique in the oven with a baking stone, but once the heat is off I don't imagine it will be much different from a preheated heavy dutch oven, with the heat off.
And finally, you might have some luck with recipes for campfire baking or other dutch oven recipes - rumstcho is correct that you can't do the kind of campfire cooking that requires surrounding the pot with embers, but there should be plenty of recipes for people that rely on portable gas burners and dutch ovens (I found this post on baking with a dutch oven on a gas burner, with emphasis on upside down cakes, cobblers, and other wet on the bottom foods).
You might consider, alternatively, looking into tagines (with something solid to use as a diffuser, since it looks like you're cooking on a flame). It will let you do slow cooking, you can roast (generally roasting chunks, not whole ingredients, but the possibility is there), you can make stews, you can bake in it, maybe not full loaves of bread but at least flatter or smaller things like foccacia or stuffed breads, pastries, things like that. I've had success with focaccia and stuffed breads up to an inch thick, which will give a nice texture more like loaf bread than flat-breads or fry-breads, and can be split for sandwiches. I will say that tagines are supposed to heat and cool slowly, and so it takes a while to cook in one - that suits me just fine, I can have something cooking in mine while I read, or do other things, and only need to check it occasionally - but you will want a regular pot and pan for when you're looking for quick food.
Perhaps the “Wonder Pot” is what you're looking for. I have no personal experience, but it is described on Wikipedia:
It was popular in eastern Europe, Germany, and Israel during periods of austerity when most people couldn't afford ovens. It is described other places as working well on gas cooktops on camping stoves and in RVs. There is a heat diffuser, which apparently comes in different sizes to match the burner, and which funnels heat up through a cooking/baking chamber that looks a little like a bundt pan, and a lid. According to Wikipedia, “A Wonder Pot can be used to bake cakes, casseroles, rice, potatoes, apples, meat, and chicken.” It's available online.
If you love braised, stewed, pot-roasted applications, a good Dutch oven might be the best $30 you'll ever spend (that's Lodge, not Le Creuset). However, if you want more variety, rumtscho's recommendation of a toaster oven is good advice. If I were going to run off in a motor home, I'd hope to take a Dutch Oven (Lodge), a toaster oven, a small non-stick skillet and a large, professional type (all-clad or similar) saute pan. Forks would be optional.
No a dutch oven won't allow you to try any and all baking recipes like a real oven will, but there are a few things that it can bake with just a gas burner, like bannock and flat breads. If you come up with one or more good recipes that you make all the time that become staples in your personal cuisine then hell yeah it's very definitely a good thing to have. Bedouins make the same kind of flat bread all the time and don't seem to mind ;) It's also awesome for deep frying. I keep a small dutch oven in my VW van and fry plantains and sweet potatoes all the time. I'm going to start making more bannock again with it and even try to do somethings with sourdough, just simple stuff like the bannock or a flat bread, nothing like croissants or a boule loaf. Fry bread is another option. I hope to try to make deep fried calzone with it if I get the sourdough working. Ham and ricotta, yum! And yeah I eat a relatively high fat diet and have lost like 50 lbs since I started 3 or 4 years ago, so don't give me any guff about the frying :) On the other hand I wouldn't use it for an acidic stew since that would leach a lot of iron into your food. I suppose that's not so bad if you give blood regularly though....