Thermal infrared would do it. (Lower the albedo, the more blackbody radiation is emitted) The JWST would be great for comet spotting, but unfortunately it's going to be used for stellar observation instead of sky surveys. Putting another ten of them in orbit would be great, but so far the JWST has taken 25 years and more than ten billion dollars to build. Infrared also won't spot a comet until it's far enough into the solar system to warm up.
Radar is tempting, but the tyranny of the inverse square law limits its use. (Worse-- the pulse has to make it out to the object, and then be reflected back. Inverse fourth power!) All those great radar images of asteroids made by Arecibo were done during their closest approach to Earth. This[1] claims its useful range was "~2 lunar distances". This was with a 305 meter dish and 1 MW of transmit power! Also, Arecibo collapsed, and nobody seems to be in a hurry to build a replacement. (The 500 meter dish in China doesn't have a transmitter)
Assuming you have infinite power, (With a moon-based dish, maybe, otherwise your gigawatts of power just ionize the air in front of the feed horn) you have other practical problems. Bigger the dish, better the gain, narrower the beam width. Arecibo had a 2 arcminute beam. 60 arcminutes to the degree: 21,600 arcminutes to a full circle. Assuming you're surveying out to Jupiter's orbit, 43 minutes for the pulse to get out there, 43 minutes back: 645 days to survey a strip 2 arcminutes wide and 360 degrees long. Which... wouldn't have caught C/2020 F8 (SWAN), which came in 110 degrees off the ecliptic. Surveying the entire night sky would take 6,074.9 years. Just out to Jupiter! You're not even looking into the cloud of outer objects, just spotting infalling ones.
