Order these in desirability:
1) Not travelling 2) Travelling by car for 2 miles 3) Travelling by car for 20 miles 4) Travelling by car for 200 miles
I've already done it. In no world is it better to own a car and travel 200 miles than it is to own the same car and travel 20 miles. In all worlds it's better to not travel at all.
You are saying that "you shouldn't drive 3 miles". Unless you are also saying "you shouldn't drive 30 miles" and "you shouldn't drive 300 miles" you're being a hipocritte.
The reason I'm such a fan of taxes on externalities is to stop micromanagers saying "you shouldn't do that specific journey in that specific way", and instead give financial incentivies to do the right thing.
If the lifecycle of a car that lasts say 150,000 miles costs $5k in externalities up front and $15k in externalities, then charge those externalities -- $5k on the purchase price, then 10c per mile.
Even in a situation where the first 2 miles costs more per mile (because in an ICE the engine needs to warm up), then charge 20c for the first 2 miles and 10c after that. Someone doing 2 miles will pay 40c, someone doing 20 miles will pay $2.20.