What we need is to fly a telescope out of the system and turn it inwards, we could probably catalog a lot of KBOs by catching them transiting.
That's a terrible idea as most of the objects are not in the plane of the solar system. And they have orbits on the order of centuries, if not millennia. Here is a better idea. We take one of the pristine mirrors that the NSA/NRO gave NASA and use them for a wide field sensitive infrared survey telescope. We place that telescope at the L2 point between the sun and the earth (where the James Webb telescope will be) as it is in shadow, cold and the earth blocks the sun most of the time. This helps keep the telescope cold so your coolant lasts longer. Then, image the whole sky. your goal is to take an image of a part of the sky with a sensitive camera in the IR bands every 5-10 days. Everything that does not move? that is a galaxy or a star. Everything that moves? Get a visual scope on that object ans see if it is real or an artifact. This has a few advantages. One the telescope can be lofted on a Falcon Heavy. Two, we know that environment and can engineer for it. Three we can set the spacecraft up to be serviced as that is about the same delta-V as a lunar orbit insertion. Three we are looking exactly opposite the sun which is where these faint objects are brightest in the sky (note we are talking 10-100x dimmer than Pluto.) Why Infrared? These objects are small, dim, faint and hard to see. But they are warm relative to the background of space and all but the brightest galaxies and close gas clouds. Some of these objects are roughly as reflective as fresh asphalt (they reflect 5-9% of the sunlight that hits them) but the get 'warm' from the sun's energy and that makes them glow like a dull ember. Most of the rocks we find are not much bigger than mountains, most are under 50Km. Oh, cool video time. Want to watch what technology can do? Here is a video of all the asteroid discoveries from the 1970's onward. Note how the sweeps occur away from the sun. In 2010 an older infrared telescope was used to look at a 90° angle to the earth/sun line and fond thousands of rocks. Those are the big events and they stick out once you see them.