ON most modern cars the brake lights work no matter the position of the ignition switch... Now.. there are several tail light hookup methods used on cars and all of them have one thing in common
The "Switched" side of the brake light switch is not powered via the RV.
On my current towed (Add a lamp system) the Towing lamp and the driving lamp. different
On my old they used same lamp but. Diode isolated.
So you find the two wires on the brake light switch. one is hot one is not (unless you are pressing the pedal) you hook a lead to this and put an indicator either on the front of the car or on the dash (Switched if on dash)
Now there is one system (M&G air/hydraulic) this wont' work with but most all the others incuding the one you have either PULL or PUSH the brake pedal. thus activating the light.
Home was where I park it. but alas the.
2005 Damon Intruder 377 Alas declared a total loss
after a semi "nicked" it. Still have the radios
Kenwood TS-2000, ICOM ID-5100, ID-51A+2, ID-880 REF030C most times