The goal is to get ~1mm accuracy
Wavelength is determined by speed and frequency. Speed is approximately 340 m/s and therefore wavelength is 8.5mm.
So what you may ask. Any standing waves you might get will occur every 8.5mm and these could ruin you expected accuracy of 1mm.
You may then point out that you will use a pulse driven into the transducer. The 40kHz resonators I've come across are very "resonant" and generating a pulse may not be that easy.
I'm saying these things because I think you need to take them into account.
A narrower beam seems logical to me or else there could be several reflections from different objects coming back and obscuring your desired distance measurement. Also remember that narrow beam devices can still produce/be susceptible to side lobe interference.
As for your other questions I think you need to determine what you want to transmit before you think about signal processing.
I'm not a radar expert by any means, but I think I understand the general concepts well enough to try to answer your questions.
What specific requirements on the peak and average powers and the widths of radar pulses was chirped-radar designed to overcome? Were these purely 'internal' concerns regarding the electronics, or were there external goals and restrictions that were hard to meet otherwise?
The basic problem in radar is to get both adequate power for total range and good timing resolution for range resolution. It is hard to build high-power amplifiers for microwave frequencies. You want to have a lot of energy in each transmitted pulse, but you also want to keep the pulse short. The solution, as you have found in optics, is to stretch the pulse by chirping it, which allows the power amplifier to operate at a lower power for a longer time in order to get the same pulse energy.
Now, in radar, it doesn't matter if you don't compress the pulse again before feeding it to the antenna — the chirped pulse works just as well as the compressed pulse in terms of detecting objects.
In fact, you gain additional advantages when the reflections come back, because now you can amplify the chirped signal in the receiver (getting some of the same advantages as in the transmitter amplifier regarding peak-to-average power), and you can use a "matched filter" to compress the pulse just prior to detection, which has the additional advantage of rejecting a lot of potential interference sources as well. The narrow pulses coming out of the receiver filter give you the time resolution you need.
Is the name 'chirped pulse amplification' ever used in a radar context?
Generally not, because amplification isn't the only reason that chirping is used.
Is the optics-style CPA - stretch, amplify, compress, and then use the pulse - used at all in radar applications, or in broader electronics fields?
Not to my knowledge, but it would certainly be feasible.
Best Answer
There seems to be a fairly large number of papers on the subject of pulse compression in medical ultrasound according to Google.
The main reason to use pulse compression (ie using chirps) is to increase the average transmitted power to increase SNR but it does come with its own set of limitations, such as increasing the minimum range response and ambiguities in the presence of doppler.
It is used with radar because the available amplifiers that can provide high-quality output are limited in power (especially with semiconductor PA) but even TWTs can't provide the peak power that magnetrons do. Magnetrons however can't provide the signal quality needed for sophisticated beam-forming and don't integrate well with modern electronics.
If the transducers can provide adequate SNR without using compression, there is not much reason to use it.