$5: probably doable. An esp8266 dev board goes for $2 on AliExpress. That leaves $1 for plastics and $2 for a lipo battery.
Was the original/intended purpose to always sell them at cost and make the profits up through increased sales?
I think in Amazon's case it was just experimentation with a sunk cost. Based on the discussion above it looks like it's impossible to form a company to sell these to turn a profit, at least in the dash button form.
negative profit when you count assembly.