If you want to run both the motor and the battery from the same 12V supply, then at the very least the motor should have its own power and ground cables between the it and the power supply. Do not share any part of the motor power or ground lines with the power feed to the video camera. But this probably won't be sufficient, because switch mode power supplies don't react instantaneously to large changes in load current.
1000uF is probably nothing like large enough to supply the current peak when the motor starts up. 20000uF would be more like it.
Consider adding slow-start to the motor. This will reduce the current surge when it switches on and give the power supply time to react. One way is to replace the switch with a mosfet, and feed the mosfet with a slowly-increasing PWM signal. Don't forget to put a diode across the motor if you do this. Similarly, use slow-stop.