One solution:
Install Snow Leopard on a USB drive, using another Mac. You could use either of your other 2010 Macs.
Update it to current version of Snow Leopard system, using the usual Software Update.
Now, boot your "new' MBAir from that USB drive.
Then, erase your MBAir internal SSD, using Disk Utility.
Still in Disk Utility - Use the Restore function to restore your external drive to the MBAir SSD.
And, that will all take a couple of hours or so, but it will work.
Last, Select your newly restored internal SSD in the Startup Disk pref pane to make sure that it is set as the boot drive.
(Once the system is installed and updated - even if the initial install is from a model-specific installer, the updated system will boot any compatible Mac. It will work in the MBAir, too.