Hiked it several times, never driven a car up it...
But, I would want a strong cooling system (including electric fan) going up and good brakes going down.
You'll be going slow up and down...way slow...
You must stay in 1st on the way down...no brake system could take the constant heating of that road...
And at roughly 6,000 feet, there is not a huge difference in air density...if the car would do OK in Denver, it will do OK at the altitudes of Mount Washington. The real issue is grade: it's steep going up and steep coming down...