Instead of a Pi you can use an Arduino with a RF module. The RF transmitter modulates at a standard frequency (e.g 433.92MHz), so the Arduino only has to send the code (a few bytes) at an appropriate frequency.

Here is an example I wrote, where I use this setup to turn on and off a remote controlled socket: https://github.com/andmarios/arduino/blob/master/lidlRemoteS...

Total cost could stay below $5 if you buy cheap components from eBay.