I pull my 1977 25ft Tradewind with a 2004 Yukon XL. It pulls it perfectly without any issues whatsoever. Mind you I have only towed it on flat grounds so far, no mountains at all.
With that said the weight of your proposed trailer is MUCH greater than mine (mine is around 5-5500 pounds), so that is not an apples to apples comparison. But I think I would feel comfortable with your proposed scenario myself if I took a few things into consideration:
1. Keep it slow. Don't be in a hurry to get anywhere. Plan an extra day (or two or three) for traveling.
2. Watch your temps, especially your transmission temperature. If you're vehicle is a 1500
series which I am assuming it is, you do not have a transmission temperature gauge but you DO have a transmission temperature sensor. You just need to get some way to read the data from that sensor. I use a ScanGauge II. Keep your transmission temps to below 190, and preferably below that (170-180 would be good.)
3. Along with the recommendation above, get a transmission cooler installed. As has been said in this thread previously the transmission is the weak link in these vehicles. They're not bad - but if you are towing more weight than what you are supposed to (you will be) then it is going to put even more stress on the transmission (and engine and suspension and...)
4. Know that you will be, technically, over your capacity. Just something for you to think about in regards to insurance, liability in case of an accident, and your own personal "comfort" with the whole situation.
5. Keep your brakes in top shape. I can not stress this one enough. I have no doubt whatsoever that this engine is more than capable of towing your rig. I have slightly less confidence in the transmission but I believe you'll be ok there too. But the brakes...you will need to get them inspected and repaired/replaced/tuned up often. This is not to scare you - you actually will probably (hopefully) never need the full capacity of your brakes as the Airstreams brakes will actually help you to stop. But what happens if the trailer brakes go out? You will need to be able to stop the whole rig with your vehicle brakes only. Using the advice above about going slow (and I didn't mention it but also make sure to leave lots of space in front of you) will help in this area a lot. But it's best to be prepared for a worst case scenario. I towed my trailer home about 80 miles the day that we got her without any trailer brakes and with brakes on my tow vehicle that were 110,000 miles old. I didn't have any problems though I am very glad that I didn't have any emergency stops that I had to make. It can be done - but it helps if you make sure your vehicle has brakes that are in tip top shape.
I was at Myrtle Beach State Park last October and was set up beside a nice couple from Canada who was towing their triple axle 34 foot Airstream with a new For Flex. It had been reinforced, worked on, etc... and I'm sure it could handle getting the trailer up to speed. My concern with that scenario would be the ability to stop the rig if the trailer brakes ever went out.
-And for my second lastly-
To top it all off - I hate to say this but you really won't know for sure until you actually get out there and tow with it. My gut feeling is that for occasional towing you will be fine. But know that you will be over the limits (possible legal ramifications?), you'll need to keep it slow, and you might end up having to do more work to your tow vehicle than you would otherwise.
With all of that in mind it is obviously up to you. I think I would do it myself but I would try to stay away from the mountains. And if it turns out that you are towing more than you had planned on you will be best to get a mid 2000's 2500 Suburban or Yukon XL (I like the 2000-2006 model years myself).
Good luck and have fun! Let us know what you end up doing.