Wiring a traffic light as a brake lights
Something like this would preserve the traffic lights as much as possible.







simulate this circuit &ndash; Schematic created using CircuitLab

A few comments. 

The inverter can be the cheapest, crappiest you can find as long as it's reliable. You aren't driving a computer here.

The relay should be a 12-volt coil with contacts rated for 120 volts at several amps. You simply connect the coil in place of the brake light bulb.

It's entirely possible that the LED modules (one per indicator) are not actually capable of running at 120 AC. At the least, it's probably rectified in the traffic light controller box. In this case, you would need to put a full-wave rectifier somewhere between the inverter and the LEDs. It's even possible that the LED modules, in addition to being DC, are not 120 volts. But you need to determine this yourself.