That's another cool solution, but it's more hackish (since you're doing it at the network level). Too bad the Dash button isn't more hackable, it's a very nice device.
Technically, Dash is a different product from Dash Button. The Dash is a barcode scanner/voice device for Amazon's grocery business (Amazon Fresh). You say a product's name or scan its barcode and it adds it to your cart on Amazon Fresh.
My problems with the flic is that it's pricy, I don't know how open it is, and I don't know whether it requires a bluetooth connection with a nearby phone.