Sorry for the late reply here. I had a similar problem last year and spent many nights trying to figure out what was causing it. I first noticed the error when practicing lunars. I found that on multiple nights, my lunars were up to 1′ off from what Stellarium was predicting, as well as what I was calculating using the nautical almanac. 1′ is a large error for lunar distance measurements, and I was determined to improve my accuracy. I was using the Astra III Pro sextant, a 7×35 scope, and was averaging 10 sights per lunar to obtain my final angles. So I was fairly convinced that the errors were not just due to sloppy measurements or bad equipment (also, I plotted up all of my measurements each night and they were always within 0.2′ of a best-fit line, so my measurements were precise, but not accurate). I had also done each of the necessary sextant error corrections beforehand, and my index error was consistently 0.8′ on the arc (which I corrected for), so I’m sure that was not the problem. What was really bothering me was that on some nights, my averaged lunars were consistently too large, and on other nights they were too small.
I decided to do some tests using star-star pairs to see if I had the same problem. Turns out, I did, but to a lesser extent. Some star-star pairs were measuring up to 0.4′ too large (relative to Stellarium’s refraction-corrected predictions), others were too small, and some were spot on. So I began systematically trying different things to figure out what was going on. I started paying attention to which way I had to rock the sextant to measure the star-star angles. I thought maybe rocking it far to the right or left would cause some very small flex in the sextant that might cause the error. After all, the sextant is (primarily) intended to be used vertically, so maybe some flex is to be expected when rotated sideways for lunars and star-star pairs. What I found was that when I rocked the sextant to the left, my index error would increase up to +0.4′, and if I rocked it to the right, the error reversed to -0.3′. But there was some scatter in the data and trend wasn’t straightforward enough to convince me it was the only thing at play. So I started looking at my errors over the course of an observing session, thinking maybe the index error changed over time as the sextant responded to the change from indoor to outdoor temperature. I found that my error would change by around 0.2′ over the course of 30 minutes, but this was nowhere near enough to account for the 1.0′ errors in my lunar distance measurements. So unfortunately, after plotting hundreds of measurements, from many nights, between multiple star-star pairs, and trying to systematically figure out where my 1′ lunar errors were coming from, I didn’t reach a compelling conclusion. But I still think it was valuable to plot the data and quantify the scatter in my measurements (I feel a little better about myself now, haha).
I did consider your point about whether or not the objects were in the center of the field of view, or off to the side. But rather than try to test the impact, I just tried to keep everything as centered as I could for each measurement. Could be worth looking into though…maybe I wasn’t as consistent about that as I thought. At this point, I’m blaming the 1′ errors in my lunars on uncertainty in the moon’s position from the almanac (stated at 0.3′) in combination with the errors from rocking the sextant to the left or right (up to 0.4′).