If it will work, you should be able to get by with ATV 3.
The only reason it wouldn't work is if DirecTV is capable of preventing AirPlay sending the signal to the ATV.
I just tried this using the native Videos app on my iPhone. Works with no problem. However, attempting to use the Flixster video app over AirPlay does not work. The video plays on my phone but will not play on my AppleTV. In fact, it appears to lock it up. I was half expecting problems with Flixster. Playing movies on the Mac version is fine on the built in display, but attempting to move the window in which the video is playing to my second display results in an error message "Unauthorized display" or some such.
Sorry to be noncommittal but it really depends on the developer of the app.
I may just visit the Apple store with my iPad and see if they will let me try it there.
Depending on the DirecTV equipment you have, you could stream live content and even download recordings from your DVR (requires Genie HR54). The screen mirroring from your iPad to the Apple TV device in the shop would allow you to display the content on the TV that the ATV is connected to.
What is it that they won't install out there? A sat receiver box? What isn't grounded correctly? Get them to provision you a wireless Genie Mini and move it out to your shop at your convenience (have them set it up on a set inside the house).
Getting the Genie right now is not economically feasible right now, they want a lot of money to convert my system at this time.
DirecTV would not install a dish out there because of their grounding requirement. Dish must be grounded to the house's main ground point. That point is at the house and not the shop. When you run power from the house to an out building, you also have to run a ground wire that is grounded at the main panel in the house. This way, everything is on a common ground and you won't have what's called a floating ground, which is against the Wiring Code.
The installer also said they couldn't just use a separate sound for just the dish and the shop. They used to be able to do this, but when AT&T bought them out, they instigated very strict rules. He told me that they actually had to submit a picture of all new dish installations showing the ground.
With our current plan with DTV, we can upgrade a piece of equipment every two years at no cost and we still have a year to go. Will go with the Genie at that time and go with a wireless receiver in the shop.
Hope this all made sense.