Radiant heat issues
Interesting idea. I've actually seen radiant installed in some very high end ($300,000+) motorhomes. But there are practical reasons why you might not want to put it in your Airstream.
First of all, let me note that I am not a heating expert, just somebody who has installed radiant in his own new home and done a bit of research on the household systems available. I'm extrapolating from that research, which may not be a valid thing to do, but I think it's at least a good guideline.
The major problem is that radiant is fairly slow heat. This is fine in household applications where you have a lot of thermal mass in the house to absorb the warmth. Think of thermal mass as a battery for heat. The bigger the battery, the more stable the temperature in the house stays, even when the system is off. For example, we put a 1" concrete thin slab in our first and second floors to absorb and distribute the heat evenly so that the house stays comfy constantly. So, if someone leaves the door open and all the warm air exhausts, just a moment after closing the door the house feels warm again because the thermal mass (floor slab, furniture, walls, etc.) is "battery" is still warm. The downside of thermal mass is that if we let the house get cold, it can take hours to re-heat that slab.
Trailers don't have much thermal mass because of their lightweight construction (insulation doesn't count -- it's to stop heat losses, not absorb heat energy). So you'd get the worst of both worlds: slower heat than with a warm air furnace, and little protection against temperature fluctuations caused by someone opening the door.
The nice thing about radiant is that once you get the house to temperature, it stays there with very little fluctuation and there's very little heat stratification (e.g., you don't just heat up a lot of air which then quickly rises to the ceiling). The radiation from the floor warms people and objects near it, which is ideal. I highly recommend it for houses, but the need for radiant is questionable in a trailer.
To install radiant, you'd need a hot water boiler (not a furnace), piping in the floor, distribution pumps, and some control valves. Electric radiant systems would draw too much power to be used except when plugged in (and even then might be too much for some campgrounds). With propane, the need for electric distribution pumps and a boiler in the hydronic (water) system might still be too much power draw for batteries and small generators.
You'd also want to get rid of the carpet. Radiant is less effective when covered by carpet, cork, or other insulating flooring material. You could cover it with a laminate floating floor (Pergo-type) or vinyl floor material.
If you've absorbed all this information and want to try it anyway, there are companies which make electric in-floor radiant heat kits for bathrooms that are very interesting. You'd only be able to use them when plugged into shore power, but they would be quite a luxury under your toes when standing in the bath in the morning! Keep in mind you'd have to put a timer on the system so it warms up a couple hours before you get out of bed.
Do a web search for electric radiant heat and you'll find some of these kits. They could be do-it-yourself installations if you are very handy with electrical stuff, and for a floor space the size of an A/S bathroom would probably cost only a couple hundred dollars.