Yes, I did mean Finity's End.  And I agree that there are three James Roberts as you've noted.
Interesting bit, though, p. 73 in the hardcover version of AU - on arrival at Pell, conversation between Emilio Konstantin and JR - Emilio says, "Captain James Robert, is it?" and then a bit further on in the passage:  "Abrezio had saddled him with that history-charged name that he pointedly did not go by - tagged him on their last night on Alpha, and his own family had picked it up, for all the wrong reasons.  He was the second captain of that centuries-old name... . Captain James Robert Neihart:  Gaia's first captain ..." 
So somehow he's JR all the way through AU but he's James Robert to all the characters within the story. 
And even more confounding, a bit further on in the passage from Finity's End you've referred to which takes place at Esperance where he's introduced as fifth captain:  "These people hadn't known he existed two seconds ago - another Captain James Robert, under tutelage of the first." (my emphasis)
I think I'm getting just too hung up on details which I may be misinterpreting and really don't have any impact on the story, anyway.  
 

So this is where I get quite confused:  in Alliance Unbound JR Neihart is senior captain and there are also Madison, Bucklin, Fletcher, among many others, of course. I understand Finity’s End is years and a war later than AU.  In FE, senior captain is now James Robert, retiring from active duty but now JR is posted to fourth captain.  And there are still Madison, Bucklin, Fletcher.  Now there's also Madelaine, who is a sitting captain and is Fletcher’s grandmother, i.e., Fletcher’s mother’s mother. So are these characters the same people, with the exception of Fletcher? 
And Tom Mallory is in AU.  Signy Mallory is mentioned a number of times in FE.  If she is Tom’s descendant, has she had enough time to be “a former merchanter, former Fleet officer and bitterly opposed to Union.  She had worked with Union against the Fleet.  There was no side she hadn’t been on, at one time or another, including Earth’s.”?

Second thought:  I think it's Mallory.

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Mini-Me