Arduino high voltage analog out
It appears that you don't know much about the input port used for your feature.
Is it a reference voltage or a reference current being used to set the operating point?
Trying to use a PWM may not work, and I'd suggest is unlikely to work, and too difficult to program if you are new to it.
Given that you 'know that the potentiometer used is a 10k (linear one assumes), then the best solution is to use an I2C digital potentiometer.
They are many available and the DS3502 may be a worthwhile choice.
You can get this as a low cost easily wired PCB such as here, which also has Arduino code shown in the example.

This DS3502 IS NOT suitable for direct connection to the 12V rail in your car (even with a 17v Absolute max rating). The 12V can easily have spikes in excess of 17-19V and could easily kill the device.
You should ensure that you include some form of supply limiting.
Since the maximum current is just a few uA I'd suggest a series 100 Ohm resistor and a TL431 shunt regulator. Study this datasheet to understand the TL431.
The shunt circuit is simple and you only need one regulator for both (or more) potentiometers you replace.

Set the shunt regulated to about 15V, which is above the maximum likely from the battery (13.8V) when not charging. This prevents the regulator sinking current on a fully charged battery and it will only conduct when the voltage is above 15V.