I suggest that you connect the camera negative to the car body, not to the light cluster, and see how that goes.
You haven't said what car it is, but most cars have light clusters where the whole cluster shares a single ground wire. Brake lights, whether incandescent or LED, will often take brief surge of current when they turn on, and that current will go through the ground wire. The ground wire will have some resistance, and so the negative voltage at the cluster will be a bit above ground. When the surge of current happens, the negative voltage on the cluster will be a bit more above ground, leaving less voltage for the backup camera, and that could be enough to make it turn off.
Having a separate ground connection to the backup camera will avoid that problem.
If that doesn't work, it could be that the car turns off the backup light briefly when the brake comes on. If so, I suggest fitting a diode and capacitor, like this:-
The diode should be a 1N4002 (or 1N4003, 1N4004 etc) and the capacitor should be 10,000 μF or more, rated at 25 V or more.
A capacitor like that is quite large, like the size of a C-cell battery so will need mounting somewhere. The diode is much smaller. Both components need to be fitted the right way round. The capacitor will be marked with a white stripe on the side where the negative wire is. The diode will have a band showing the cathode (labelled K on the diagram - I don't know why diode cathodes are labelled K, but they are).
When in reverse, the capacitor will charge through the diode, and run the camera. When the voltage to the backup light stops, the capacitor will run the camera for a short time, and the diode will stop current going back to the backup-light. The capacitor will only be large enough to run the camera for a few tens of milliseconds, but I think that is all you need.
I don't know if that will work, as I don't know the exact cause of the camera cutting out, or how much current the camera takes, or how low the voltage has to go before the camera cuts out, but it's probably worth a try.