You basically do AM of the laser, and then otherwise normal baseband radar processing. You don't need much bandwidth, but more bandwidth needs less SNR, as you can do superresolution if you have sufficient SNR.

There is no need for an FPGA, but you will rather quickly throw FFT correlation processing at it.

This would not use any delay lines. You'd just need to lock the sampling of your ADC to the signal generation, because 2ns jitter are 30cm/1ft jitter in distance.