Microwave sensor beamwidth is dependent on antenna size and wavelength.  You could go for MMW sensor at 35 GHz and with 6" diameter antenna get a 16" spot at 10 meters.

Laser can be used.  Just a question of wavelength, filtering and thresholding.  BUT, commercial lasers with 1 kHz sampling are available with ~3 mm accuracy:
http://www.sickusa.com/Publish/docroot/Technical%20Information%20Sheet(s)/DME%205000%20Technical%20Information%20Sheet.pdf

Your parameters indicate less than 0.2 s for total measurement period, so <4 shots at the PRF you've specified.  At 1 kHz, you can get nearly 200 hits, to be used for range correlation/noise filtering.

TTFN

IRstuff is corect. Your comment that microwave ahs too large a beam is incorrect.  To get a good sample rate output of 20 Hz you might actually want to sample updates much greater than that, and synthesize a 20 Hz estimate update.  YOu could have tremendous triple transit or mutlipath problems.  Since you are looking for 3 cm you might have to code your signal and get some signal processing gain.  You would also get a radial speed estimate for free, since you will probably have a tracking filer and velocity estimator in order to corect average your distance numbers.

Laser is probably best, with microwave probably second.

But if you end up back with ultrasonic, then you could have multiple pulses in the air at once to defeat the apparent transit time limit. To ensure that your return isn't swamped by an outgoing pulse, you can either ensure high T/R isolation, or cleverly vary the spacing on the outgoing pulses to allow the returns to sneak back in between at one point or another (requires full analysis to optimise the timing pattern). You might have to also code the pulses to distinguish which is which when there are two or three in the air at once.

Laser would be so much simpler.  Police speed measuring laser systems work by measuring distances over time (not doppler shift as with police radar). They can be operated over relatively huge distances (as compared to 10m). Of course you code the IR laser ("AC") to futher distinguish it from sunlight ("DC").

Microwave should be quite simple over a 10m path (20m return). Use a swept microwave and the tone (frequency delta between what is being emitted now and then, then being after the return) can put the distance into the audio frequency domain where you can easily measure it. Watch out for sideways movement causing doppler.

3cm is fairly tight... You need to decide exactly where on the car you're measuring from.  Most cars have more than 3cm of surface variation since they're not slab sided. Maybe you can install a corner reflector in one window?

Another problem with ultrasonic's transit time is that the car is only 'there' for perhaps about 100ms (at 100kmh).

The Speed of Light would be quite handy.

At 64 GHz the two tone RF method is somewhat standardized on, but not shipping in any product yet.
Yo would use your chirp to give you the round trip distance and the doppler can help with relative perpendicular velocity.

CW radar altimiters designed for helicopters and low altitude aircraft, and now used on cruise missiles, give a continuous accurate range readout at distances to < 3m.

These are available from several manufacturers, especially the European ones, but the key is CW radar, not pulsed.