It's about being hard? No, clearly it's about being cool. If you lose your cool, you'll panic and swerve. If you're able to just keep driving straight, confident that the other car will swerve, you're damn cool. Roll to act under fire, with the fire being "you turn before they do".
Regardless of whether the Driver chickens out, will the NPC? Will they chicken, or do they have enough of a death-wish to crash if the Driver stays on course? Or will they stay in for longer than their driving skill can handle, thus crashing while trying to swerve? You as the MC have to make this call, I think, but you can use any of the handy options to disclaim decision-making. With this information, the cool roll above is clear. Perhaps you want to be generous with the truth, and tell the Driver "it's no use, Tum Tum is crazy, they'll just crash into you" if that's the case.
If you decide they would swerve "at the last moment", though, then a cool roll would be great to determine whose "last moment" is actually last. 10+, Tum Tum chickens out. 7-9, maybe they crash from being overconfident (you prove your cool but they do too, to grievous bodily harm for you both), or Tum Tum swerves but the Driver also wavers. Miss, maybe the Driver panics and swerves first, maybe Tum Tum or an accomplice draws a gun and starts firing at your windshield while you're at it.
Alternate ruling: you're threatening them, but you don't actually intend to follow through. Roll to manipulate. Your promise is that you'll keep driving, and the concrete promise is... driving so fast and furiously that it warrants an acting under fire roll not to dent their car while passing, perhaps, or just to not crash after passing them? Or, I dunno, locking the pedal and climbing on the roof. That's concrete assurance you're committed (if you don't want to die you can still "chicken" by jumping off, but you have staked your car).
If you really think it ought to be hard, write a hard custom move with the outcomes you envision. Maybe they are you swerve / you crash / they swerve, or maybe they're something else. I'd say it depends on the NPC. With a crazy enough NPC, it's not about whether you're hard enough to stay in; it's whether you're sharp enough to realize they have a death wish, and chicken out in time. Imagine playing against, like, any of the characters at the end of Fury Road. Rictus Erectus would crash a car and die to bring a halt to the war rig, as would any other war boy.
Being a fan of the player's character, however, means that if there's no reason to believe the NPC would stay in to the end, a successful cool roll means the NPC chickens out.
Finally, if you really want it to be not only hard but go aggro, I think exchanging harm on a 10+ makes sense in this context. Imagine you're going aggro with a grenade at close quarters, like Leia (disguised as a bounty hunter) in Jabba's palace in Return of the Jedi. If they force your hand, they take harm, but your weapon of choice happened to have a hell of a recoil. If you can go aggro with a messy weapon, hurting people other than the one you threatened when you fire, then I think you can go aggro with a car, hurting yourself when you "pull the trigger".
I think the cool roll is more appropriate, though.
Fake edit: yeah, derendel said much the same things as me.