I think you want 3 states. RED LED, Green LED and no LED. Where you are going this can not happen with out more parts.
First your circuit can only make one LED come on. I changed the way the LEDs are wired up.
Next I used R3, R5 to divide Vbat down to about 1/2. (near 6 volts)
I added D3 to make a good Vref.
I used only one supply V2 so it is more like what you will do in a car.
I do not have the op-amp you do so I just picked on at random. Could be really wrong.
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For simulation I made a battery V2 ramp from 9V to 13V so you can see what is happening.
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I know this is not what you want but it is a start.