Thank you all for your help and suggestions. I think pdentrem is on the right track with the leadscrew moving left and right. I took off the one and only “thrust bearing” surface (or “adjusting disc” per Tom’s comment) and indicated the thrusting face on the leadscrew to find...0.0045” of variation (see attached video). That’s not to say that face itself is responsible for all the variation, as the opposite face may have some variation as well (since I was pressing the leadscrew left while rotating it). It also could be interference from the key at the far left end of the leadscrew pressing against a washer unevenly. I took the gears, key, washer, and bushing off the leadscrew to take the measurement just to give a bit more control. Note that both the indicated surface and the possible key-washer interference would be surfaces under force when cutting threads right-to-left (the leadscrew being “pushed” to the right by the carriage drag), as I’ve been doing for all the tests.
So, I think the next step is to take the leadscrew to a more capable (longer) lathe, set it between centers, indicate it square, and clean up the thrust surfaces. I’ll clean up the “adjusting disc” as well (it’s a pretty rough cast iron piece) and put in a bronze washer on both left and right thrusting faces to get smoother motion and reduced wear. I’ll post an update as soon as I know if all that works, but will surely be listening in for more great advice.