I do a fair bit of storm filming (mainly UK based but not always) and a lot of other stuff for family members etc. If you use a directional mike externally it will likely be just as sensitive but in a narrower 'beam', so positioning to get exactly what you want could still be a nightmare. Road noise from tyres, other vehicles, ambients, mechanical vibrations, wind noise - all could give you a potential headache. Internally it might work but you'll still have similar issues.
One alternative is in the edit. Assuming a half decent edit package, you should be able to tweak the frequencies to bring up the low end notes such as engine growl, and drop some of the high end stuff such as shrieking passengers. It's an art not a science as such but I've found it useful on more than one occasion.
Option 3 is fake it. Build up a library of 'clean' audio to your spec and edit it over the original footage, adding passenger comments that fit as appropriate. Most half decent packages allow at least two audio streams to be edited and some many more.
Option 4 would be to have a word with passengers, tell them what you are looking for and 'sell it'. to them. Most will play along quite happily. I do this when I'm out chasing. When the camera's running, anything not related to the event is (politely) discouraged. Anything else could be considered a natural reaction and part of the experience. What you seem to be talking about is a planned shoot, rather than an 'as it happens' record so it's not really out of the question.