Possibly either of these would be suitable for the Pi -
Plus either a tiny step-up or a separate skeleton type psu for the 12V output?
For the controller, the ESP8266 or similar type modules are apparently very capable, with built-in WiFi and a programmable MCU - some people on here rave about them, but I've not used them myself.
Amazon.com: ESP8266 microcontroller NodeMCU Lua V3 WiFi with CH340G : Cell Phones & Accessories
www.amazon.com
The power required for one of those is trivial compared to a Pi & they are a fraction the size & cost.