You are describing the end-user price of a device that includes batteries, a circuit board, a WiFi transciever, a switch, and an enclosure. If the BOM and assembly costs can be brought down to $4.99, that's a miraculous feat of engineering and modern civilization's economies of scale; if it can end up on your doorstop for that price it must be subsidized.
How else could that device cost less than the gallon of industrial chemistry that you order with it?
Anyway, here's a teardown I found:
https://mpetroff.net/2015/05/amazon-dash-button-teardown/
Now a surprising thing that I didn't know before - this thing has a microphone on board. Yes. Yet another Internet-connected microphone. Apparently used only to configure the button via ultrasound, but I wonder how hard would be to repurpose it...
$2.72 for a single programmable WiFi device, delivered for free. All you need after that is a cheap battery, button, and enclosure. The existence of the ESP8266 is changing the way I think about electronics.
which is subsidized by the USPS and Chinese government using mail reciprocity agreements to ship for pennies.
https://www.washingtonpost.com/news/storyline/wp/2014/09/12/...
This way you will have months or even years in autonomy instead of weeks.