Voltage isn't used to dim LEDs (they are current devices).
But chances are that your vehicle's dimmer is a PWM type unlike old types that were rheostats (big variable resistors - aka pots/potentiometers - that varied voltage).
A problem may that your backlight is (probably) powered internally from a regulated (voltage) source, hence it's not as straight forward as using the vehicle's PWM dimmer (eg, typically varies from 12V - 14.5V).
One option is to insert a PWM circuit - eg, a 555 timer with a pot and a few other components (resistors, diode(s), capacitors) - eg, see recommendations for attiny or arduino newb? Reply #23.
If you want the backlight to track an existing (PWM) dimmer, maybe a Zenor diode circuit could be used (clamps the maximum voltage), or a fast reacting voltage regulator (that doesn't mind operating at the PWM frequency), etc.
Maybe even a PIC (a $3 PICAXE 08M2 plus circuitry & program) that measures & duplicates the existing dimmer's duty cycle, or analog dimmer's voltage (LCD driver?), or modifies based on ambient lighting, or overrides based on your adjustment (ie, manual & auto modes)...