NavList:

Yes, indeed. For the position of the Moon or any object closer than about a million miles from the Earth, like artificial satellites, the actual position of the object "in front of" the celestial sphere changes by a tenth of a minute of arc or more when the observer moves by as little as twenty miles, which is the scale of position changes due to the Earth's oblateness. Really, though, this is a modification to the object's parallax calculation. 

Dealing with oblateness in lunars was never a serious concern, certainly not in the era when lunars were used at sea. Chauvenet was very late to the game, and he was a desk-bound mathematician. Revered in the history of the US Naval Academy and in the history of American positional astronomy, too, he greatly over-estimated the importance of lunars, and, as I have said before, I highly doubt his method of clearing lunars, which was given a ceremonious place in editions of government-issue Bowditch after 1880, was used more than two dozen times in real navigation at sea. It was a pointless change to Bowditch which already included excellent methods for clearing lunars. The refinement of the oblateness correction is nice to know about, and there's no reason to skip it in digital lunars computations, but navigators, real navigators back then, had no use for it.

The impact of oblateness is really very small. You can test this yourself by trying out cases using my web app (which has been correcting lunars for oblateness since it was first launched online over thirteen years ago). Go to ReedNavigation.com/lunars, select Clear a Lunar and enter the observational data for any lunar. You can pull the data from a historical textbook, or from posted examples in old NavList posts, or you can generate them in simulations.... or you can step outside in the afternoon or evening of the next few days and shoot a few! However you get your data, you'll notice that my web app has an explicit option to "Ignore Oblateness of the Earth". Normally the oblateness correction is included. That is the default. But you can turn it off to assess the impact of the oblateness in specific cases.

You mentioned that the effect of the oblateness seems to grow towards the poles. That's true enough, but don't forget that traditional lunars were used to determine longitude. And since longitude lines converge at the pole, that slightly larger error in longitude is cancelled out, and the error on the ground in terms of nautical miles actually diminishes. In statistical experiments that I ran years ago, the impact of the oblateness corrrection amounted to less than two nautical miles in the majority of cases --really not important for practical lunars back in the 19th century.

By the way, William Chauvenet sure could have benefited from the Internet back when he was excited about lunars! In one of his first papers extolling the virtues of his "new improved method for clearing lunars" (a phrase which had become nearly a cliché by mid-century), he complained quite arrogantly about a method introduced "recently" by one "Mendoza Rios". If he had the Internet, or even just a well-paid research assistant, he would have known that Don José de Mendoza y Rios was one of the true experts in the history of the mathematical analysis of lunars, who in fact scooped Nathaniel Bowditch on his first method of clearing lunars, and he would have known that Mendoza Rios had died by his own hand fifty years before Chauvenet was complaining about his "recently introduced" method...

Frank Reed