However, as far as I could see, for my sort of observing (i.e. to the nearest minute of arc), so long as the sextant is held vertically, being slightly off the line of sight isn’t a problem, and being a long way off line of sight means you can’t see the Sun.
Yes, the moving Sun would be a problem when trying to see this effect. A terrestrial angle would be better, with the sextant held horizontally, so long as the terrestrial targets are distinct, reasonably far away, and say 90 degrees or more apart (and of course stable). Or, at night, a pair of stars at comfortable elevation, say between 15 and 30 degrees, and 90 to 120 degrees apart.
Let's see. The Davis Mark 3 has a sight tube with 11 mm internal diameter, about 165 mm from the horizon mirror. So if the eye is at the extreme left or right portion of the sight tube (or telescope mounting ring; but those are generally larger, maybe 25 mm or more), then looking at the center of the horizon mirror, the line of sight is canted away from the plane of the sextant by roughly 5.5/165 radians or about 2 degrees. With a 25 mm ring it would be 4 degrees (!).
Such large miscollimations should be visible. The sextant books go through the derivation of the error, but the short of it is that a 2 degree miscollimation, when observing two objects 80 degrees apart, will result in an observed angle 3.5 arcminutes too large. That should be visible even with the naked eye at unit magnification. I haven't tried it in this exact way (though I have been bitten by collimation errors in the past), but it seems like sliding the eye from a centered position to the edge of the sight tube and back, while maintaining both objects in the center of the horizon mirror, would result in the objects alternately approaching and receding to the tune of 3 arcminutes (or four times that with a ring twice as large, which would be unmistakable).
With a telescope, the usual advice to measure miscollimation is to keep the eye fixed at the exit pupil of the system, but yaw the sextant so the objects go across the field of view. If the point of minimum angle is not in the center of the field, there is some miscollimation. That's perfectly okay---just observe at that point, not at the center.