Interestingly, we'll solve this problem just about when it won't matter anymore for New Horizons. In the 2020s the Large Synoptic Survey Telescope will come online, which will (among other things) catalog tens of thousands of KBOs/TNOs.
Isn't TESS kind of built to do the same thing? Wide field analysis? Presumably it's looking for dimming rather than relative motion, but that seems more like a software issue than a hardware one?
Not exactly. TESS monitors bright objects. It uses actually quite small telescopes (only 10cm aperture or so), but the it has 4 such telescopes and uses very wide fields of view to monitor a large number of stars continuously. To see KBOs/TNOs requires more light gathering power.