There are numerous Stick/TIG boxes for under $1,000. Often, the TIG is only a "Lift Arc" start which is less desirable under certain circumstances (instead of being a "High Frequency" start too). Also, the TIG will likely only be DC so it won't weld Aluminum. If that meets your needs, Hobart, Lincoln, Miller (MaxStar) come to mind. Many of these boxes may only be 120v, so that will limit the single pass thickness and duty cycle you can weld.
You might also look around your area at Craigslist for a used welding machine. Or, find a local Repair shop, they often have used boxes for sale too. Craigslist items would atlas allow you to personally inspect and verify the machine before you buy it used.