You should be able to get an Arduino to do what you want, and you'll learn how the circuit works in the meantime, as well as be able to do all kinds of other things with it.
Aside from that, there's been a few things out in the past but they've never worked how I wanted them to. I have an old Tamiya brake light controller that turns on the brake lights when off-power. But it runs on a capacitor that charges while it's getting power, then releases the energy stored when off-power. So they don't stay on forever, and they come on when you're not on the brakes. With an arduino controller, you could tell it to send the brakes on when it sees the right pulse range from the receiver. It'd probably be an extremely simple circuit and program, really.