[bobski]

The LCD backlight is a simple LED/resistor series circuit designed to run at 5V... It should be separate from the rest of the display electronics. #2 in your pic is the positive lead, #3 is negative. Since you want to use your dash circuit for dimming, you can disconnect the backlight from the Arduino all together and eliminate the 2N3906 transistor and 220 ohm resistor.
Pick up a 150 or 180 ohm 1/2 watt resistor and a 1N4001 rectifier diode (in case you get the wires backwards) from radioshack or where ever and hook them up in series with your backlight. Line end of the diode to your backlight positive, resistor on the negative (resistor orientation doesn't matter). You should now be able to hook it up to the lighting circuit just like your vacuum gauge... The remaining diode lead to lighting circuit positive, the remaining resistor lead to the dimmer/negative.