Welcome to the Forums!
What you propose is not impossible, but it will take some time to find both a tow vehicle and Airstream in the budget that you propose. An option would be to consider an older tow vehicle that can be had for a very reasonable price such as:
- 1990--1996 Buick Roadmaster with 5.7 LT1 V8
- 1990--1996 Cadillac Fleetwood Brougham with 5.7 LT1 V8
With the optionally available tow packages, the above two vehicles could be had with up to 7,000 pound trailer tow ratings. Since fuel prices skyrocketted, the prices for either of the above cars have fallen significantly. It isn't terribly unusual to find a one-owner that has been well-maintained with carefuly searching.
You would be a bit more limited in your trailer choice than with a 3/4-ton pickup, but there are many Vintage (pre 1985) coaches that fit within the trailer tow ratings of the above.
The critical thing if you decide to consider what would be a vehicle that is 13-years old or older having a mechanic carefully inspect the prospective tow vehicle is very important -- it may cost upwards of $200, but could potentially save you several times that amount.
I may be a bit too willing to consider older tow vehicles since I have had good luck with mine. I towed my Argosy Minuet over 3,500 miles last summer with my 1975 Cadillac Eldorado Convertible. The only problem that I encountered was a failed alternator -- something that could have happened with any tow vehicle. I will be using the same tow vehicle to tow my Overlander to the International Rally in Madison, WI this coming summer.
Good luck with your investigation!