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The LOP passed near Gardner only near the time of sunrise and moved quite far away by the time the plane could have been approaching that island. You seem to have missed the whole point of that web page.
gl
I didn't miss the point, but I think you missed mine:
Your site includes authoritative information on the use of sun lines on the "page 157" link. Despite the example used in that text being nearly verbatim the same case AE and FN would have faced, you continue to discount that approach for some reason beyond my understanding.
As to sun lines, LOPs and movement -
Once established, the LOP does not "move" with the sun. Think about it - if they did, they would be useless.
A LOP could easily have been established by offset - FN would have known how far east or west he was at sunrise by the time. From that shot he also would have had a good idea of speed. Therefore it would have been a simple matter for him to project by offset a LOP for Howland: then it would be a matter of elapsed time by which to judge arrival at the LOP.
Once at the LOP it is a matter of track - simple enough and the obvious outcome would have been 157 - 337.
Simple indeed - and foundationally sound. Given the nature of able humans like FN when confronted with a need to solve a problem, I suspect FN would have done about what Occam says is rational - and LOP is about as foundational as it gets in the case he was facing. I think that's where I'll keep my money... 
Cool link between FN and the author of that material - thanks for sharing that. Small world!
LTM -
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O.K.?
You did get the point that the LOP is perpendicular to the azimuth of the sun at the time that each observation is taken, right?
And if you haven't lived your whole life in a cave you have probably noticed that the azimuth of the sun changes during the day, rising in the east, passing through south at noon, and setting in the west, right?
So since the sun's azimuth changes during the day and the LOP is always perpendicular to the sun's azimuth then you do see that the azimuth of the LOP changes at the same rate as the sun's azimuth, right?
Then how can you say that "the LOP does not "move" with the sun"?
gl