Small Detail about Jaime Lannister and the Knight of Skulls and Kisses (Spoilers Extended)

If you are like me and believe that Richard Lonmouth = Lem Lemoncloak, then I found a nice small little detail that could have some type of effect on the showdown between the Brotherhood without Banners and Jaime/Brienne in The Winds of Winter.

One of the potential storylines in this showdown is the fact that if this theory is true, then Jaime and Richard should recognize each either. But I just realized that not only should they recognize each other, but its also possible that they don't like each other.

Nor did His Grace agree to appoint Lord Tywin's son Jaime as squire to Prince Rhaegar; that honor he granted instead to the sons of several of his own favorites, men known to be no friends of House Lannister or the Hand. TWOIAF, The Targaryen Kings: Aerys II

So we know that House Lonmouth or at least the lord/possibly heir (Richard) were men "known not to be friends of House Lannister"

Obviously it would be much stronger if we had some mention of Richard in Jaime's chapters (he never thinks of him) but Jaime also left early from the Tourney at Harrenhal (the primary place Lonmouth is mentioned).

I don't know if this really helps the theory or what, but Lem does mention Jaime on numerous occasions. That said there are quite valid reasons (his support for Beric/LSH, etc.) as to his hate as compared to previously having some contention as Richard v. Jaime:

He does seem to get overly excited here:

Lem cracked his knuckles and said, "Wouldn't Lord Beric love to capture Jaime Lannister, though . . ." -ASOS, Arya III

and interested here:

"Did they say how Lannister came to escape?" Lem asked.

"They did," said Lady Smallwood. "Not that I believe a word of it. They claimed that Lady Catelyn set him free." -ASOS, Arya IV

and angry here:

"Lannisters," Thoros said. "Roaring red and gold." He lurched to his feet and went to Lord Beric. Lem and Tom wasted no time joining them. Arya could not make out what they were saying, but the singer kept glancing at her, and one time Lem got so angry he pounded a fist against the wall. -ASOS, Arya VIII

and:

". . . before his friends cut her throat for her, that must have been," said the big man in the yellow cloak. "We all know about the Kingslayer and his oaths." -AFFC, Brienne VIII

and:

"She wants her son alive, or the men who killed him dead," said the big man. "She wants to feed the crows, like they did at the Red Wedding. Freys and Boltons, aye. We'll give her those, as many as she likes. All she asks from you is Jaime Lannister." -AFFC, Brienne VIII

These easily could be construed as just Lem's support for LSH, but who knows.

But we should keep in mind how the alternate situation turned out as well. Tywin originally intended to betroth Cersei to Rhaegar, before being rejected. We all know about what happened to Elia.

Well, Prince Rhaegar married Elia of Dorne, not Cersei Lannister of Casterly Rock. So it would seem your mother won that tilt."

"She thought so," Prince Oberyn agreed, "but your father is not a man to forget such slights. He taught that lesson to Lord and Lady Tarbeck once, and to the Reynes of Castamere. And at King's Landing, he taught it to my sister. -ASOS, Tyrion X

Using that same logic there should have remained some animosity (since there already was some) between Tywin and Richard Lonmouth/Myles Mooton.

men known to be no friends of House Lannister or the Hand. TWOIAF, The Targaryen Kings: Aerys II

If interested: Brienna's Dilemma with Jaime/Lady Stoneheart and Surviving Lady Stoneheart

TLDR: Some possible animosity previously exists between Jaime and Lem Lemoncloak (if Lem indeed is Richard Lonmouth).