There are very few run-flats on the market at present, mostly made as OEM equipment for specific vehicles. Chevy Corvette, Toyota Sienna, various BMWs, all the 2009 or later Mercedes diesels, etc. You might check tirerack.com to research particular sizes. A lot of them (perhaps all of them) wouldn't fit on an Airstream. I did a quick search and found no run-flats in the 225/75R15 size.
There are different run-flat technologies in use. Goodyear's Eagle LS-2 design uses a reinforced sidewall, for example. I have no idea if the run-flat designs are suitable for trailer use, and short of talking to an engineer at Goodyear you probably won't get an educated opinion on the subject. There would be complex considerations regarding pressure, load rating, sidewall strength, tread wear, etc. People will speculate about how run-flat designs might be suitable or unsuitable, but keep in mind that's just speculation and not fact. This is one of those cases where your customer would be the first test pilot if he/she tried it.
At a minimum, your customer would need a TPMS (Tire Pressure Monitoring System). Otherwise a tire could lose pressure and the driver would not know it until it shredded, resulting in exactly the problem you're trying to avoid. With the combination, and Pahaska points out, you'd be able to continue towing slowly for a short distance until the trailer could be safely pulled over.
However, run-flats are incredibly expensive. The Goodyear Eagle LS-2 Run-Flat runs about $400 per tire! Also, if you use the tire while flat for more than a short distance, it must be replaced. That makes for an expensive single-use spare. So even with run-flats, you must plan to stop immediately and replace or repair the tire, just as you would with a traditional tire. So the advantage of having an expensive run-flat is minimal.
Considering all that, I wouldn't mess with the run-flats. My choice is to use a TPMS with standard trailer tires, and carry a compressor and a tire plugging kit. The real problem Airstreamers seem to have is that they don't know when they've got a loss of tire pressure until too late. A TPMS is cheap insurance and I recommend them to everyone.