How can I use transistors to work as an SPDT relay using the common pin as input and NC/NO as outputs?
For me, the simplest method would be to convert the 12V input levels into TTL 5V levels. The floating inputs pulled down to zero using a resistor. Then program the behavior using a small microcontroller. The outputs could be either relais or power MOSFETs. Maybe there is such a device already to buy. If you don't have tools to play with electronics then maybe arduino? Using this method, you could even program the brake light to strobe like in Formula1 Hehe :)

  

For the output a RFP30N06LE MOSFET would do.