Got around to doing my "tilt the Kate" experiment. Loaded up the bobbins, clamped the lazy Kate to the table on a forty-five degree angle, and plied my yarn. No improvement in performance that I could tell. Running the strands through the back of a chair helped to get proper tension.
You can get lazy Kates that hold the bobbins at a forty-five degree angle with no propping or clamping necessary. A friend got a model called Kate 45 from cjkoho on Etsy and loves it.