Flight controllers usually have PPM inputs, not outputs. The PPM output is on your radio module.
You can indeed connect the PPM output signal to both the FC and your PPM->PWM converter: the FC would be controlling the motors, and the converter would be controlling the servo. There's nothing to configure in Betaflight in this case: the value from a certain channel will be applied directly to the servo.
A better idea is to configure Betaflight to control your servo: this avoids using the converter altogether, saving space, weight an money. If you have 1 or 2 servors to control, it's recommended to use
feature SERVO_TILT, otherwise
CHANNEL_FORWARDING is an option. Here's the Betaflight documentation page with examples. Note that you need to assign resources (pins with timers) to servos before you can use them.