If you are not willing to do wiring then what about simply using a cheap IR demodulator ( 38kHz type which is in a 3 pin package, often free in old electronics with a remote control already on a PCB with appropriate connector, IR window and wiring loom) looking into the car interior, and use a 38kHz IR from a IR LED to give the power signal to it. Just mount the transmitter where it can get a switched supply, and use the receiver output to control the dashcam power supply with some delay so momentary interruptions do not trigger the camera off.
Otherwise simply use a high pass filter on the 12V rail, a capacitor to pick off noise and amplify and detect for this, so you run the dashcam when the engine is running and the alternator is generating a lot of noise on the 12V rail. A lot more reliable than detecting battery voltage which will vary with battery temperature, though you can be fairly sure if the battery voltage is over 13V the engine is running, but heavy loads ( rear demister, defrost running using fan on full with AC running as well, electric seat heaters) can drop the voltage in the instrument console down below the level of 12V quite easily.
I mounted my dashcam there as well, drilling and tapping the rear view mirror pillar M5 underneath to hold a hand made mounting adaptor which put it close to the window, mounting by the standard 1/4in camera mount insert. Power supply is tucked up under the hood lining, with the little SMPS taken out of the case to put better and bigger capacitors there ( original was mystery branded, so put in bigger slightly better brand versions and put in a fusible resistor for protection) with it wrapped up in a sleeve. Power is from a wire dropped down the lining ( hardest part was getting it under the lining to the A pillar) to a supply tapped off a 12V switched line.