A low-cost option with excellent open-source library support would be a 2.4Ghz nRF24L01+ RF Transceiver Module, sold on eBay for as little as $1.90 with free shipping.
This can be controlled by a low-cost basic microcontroller, such as the popular Atmel ATtiny25/45/85, on a board with minimal support components.
Both the module and the microcontroller can be operated at 3.0 Volts using a CR2032 button cell. Component count is thus reduced further by eliminating the need for a voltage regulator. Individual unique IDs of 10 or more bits length can be stored into the microcontroller at programming time.
Add a pushbutton, enable the microcontroller's internal pull-up for the button input, and you have a truly minimal ID transmission wireless device.
This meets the range requirements in the question, and supports the ID length specified.
- Here is a comprehensive tutorial on using nRF24L01 devices.
- If you do not want to use a pre-built module, the reference PCB layout provided in Figures 33 through 38 of the datasheet can be extended to incorporate the transceiver, PCB trace antenna, microcontroller and pushbutton on a single board.