You've been given some great input.

Matt Sibole's mentioning of the monopole is worth considering. It's possible that there is some eccentricity in the pole at one of the joints. The best way to tell this would be to find a sturdy shelf about 5' above a smooth floor and drop a plumb bob from a mark on the bottom of the shelf to the floor, then mark the floor. Extend the pole till it compresses between the floor and the shelf. Calibrate the bubbles, then turn the pole 180°. If the pole is straight the bubbles should still be centered.

However, you mentioned that you maintain the orientation of the LS during the observation, which would likely cancel out any error due to eccentricity provided that the eccentricity in the joint(s) is consistent. If the joints are wobbly, then you've probably found your problem.

As Matt Johnson pointed out, from a statistical point of view, the best estimate for the* true* position of the point (which is never truly known) is the average of the two which would suggest that each has an error of 0.03'. From my own testing, I'd generally expect the accuracy (at 2 sigma/95%) to be 0.027' for each of these observations. The repeatability of two observations on the same point with an accuracy of 0.027' in each observation would be 0.038', so your 0.06' does seem a little high, but not unbelievable. (Note that my accuracy estimate does not include centering errors.)

Lastly, keep in mind that the discussion of epochs largely comes from a time before beast mode and thus epochs and seconds are virtually interchangeable. Of course, two epochs over 4 minutes is not the same as 240 epochs over 4 minutes or 1200 epochs over 4 minutes as are more data points to average. But time is still important, which means that 240 epochs at 5Hz can be done in 48 seconds but isn't going to be as good as 240 epochs over 4 minutes.

In my own testing, I found that there is a point of diminishing returns with more time on site. The difference between the accuracy of a 30 second observation and a 4 minute observation was about 10-15% (if I recall correctly). Past 4 minutes, the accuracy improves very little, even out to 20 minutes.