From Wikipedia data, let's assume that the average comet can be approximated by a sphere 5 km in diameter. Let's also assume that it's a perfect conductor, so its radar cross section is that of a 5 km metallic sphere, or 19e6 m^3. Since you're interested in observing comets in space, let's assume a range of 400 km, or low-earth orbit.
The highest EIRP one can radiate without a license in any ISM band (in the US) is 53 dBm at 5.8 GHz. Let's assume this is split up as 30 dBm (1 watt) transmitted power, and 23 dBi of antenna gain. When limited in transmit power, one must resort to FMCW radar to achieve decent sensitivity. Due to use of FMCW, peak transmit power and average transmit power are equivalent.
Plugging these numbers into the radar equation (and assuming no attenuation due to gaseous attenuation or rain fade), one gets a receive signal strength of 4e-17 watts = -144 dBm. For this to be above the thermal noise floor of a perfect receiver (no added noise), one would need a receiver with a bandwidth of ~1 kHz - which doesn't leave much room for frequency-sweeping an FMCW transmitter.
Turning this around, if we assume a receiver sensitive to -113 dBm (0.5 MHz bandwidth, with a 3 dB noise figure) then the maximum range at which you can pick up the echo from a target with a comet-sized RCS is about 21 km.
One way this can be made to work is to use lots of radars and add their outputs in-phase (interferometry). This would require synchronization between the radar nodes - not impossible, one can use GPS for this, but it would be expensive, and you would need a lot of radars to get the type of sensitivity required to detect a comet at 400 km range.
If you disregard FCC rules and set up a 5.8 GHz radar using a surplus TVRO 14 foot [4.3 m] reflector, the antenna gain goes up to about 43 dBi. The received signal would be higher in power by 20*2 = 40 dB, which is in the realm of what one can receive and process with an SDR.
As always with FMCW, the trick is to prevent the transmit signal from desensitizing the receiver. Separate transmit/receive antennas are one way of doing this. Active carrier cancellation is another, but is tricky to implement.