In order to control the camera via UART, not only do you have to find a UART to connect it to (that shouldn't be a problem, I doubt, that you'll use up all available UARTs - see https://pixhawk.org/users/wiring#serial_ports), but also the flight controller has to know how to "talk" to the camera. That is, the protocol for interfacing with the camera and the corresponding logic must be implemented in the flight controller firmware. As far as I know, PX4 does not implement it.
I know, that runcam cameras also allow controlling them in a limited fashion via PWM (see https://support.runcam.com/hc/en-us/articles/115010502648-How-to-use-remote-cable-on-Split-1-2-). You may connect the camera directly to some PWM output of your RC receiver if it has PWM outputs. Or, you can connect it to MAIN5 / MAIN6 (see https://dev.px4.io/en/airframes/airframe_reference.html#quadrotor-x) of the flight controller and feed one of your RC channels to one of those outputs.
That way you can start / stop recording and take photos using your RC transmitter.