Basically, this. There are numerous logical problems with the OP and this particular line of reasoning. None of it seems to be backed up by the novels.
The books have never stated or implied if the Others have a king or leader. It isn't unreasonable to assume that they have some sort of leader or king but nothing has been said about that. And there's certainly nothing to suggest that Rhaegar would somehow end up leading the Others. It doesn't track that a Faceless Man would be hired to replace Rhaegar. Why would a Faceless Man even agree to do that? They're assassins, not bodyguards.
It was Lyanna that loved blue winter roses, not Rhaegar. If the blue winter rose at the Wall represents anyone, it's Lyanna. And who is at the Wall who is most likely a part of Lyanna? It certainly isn't Samwell Tarly...
What we know about the Night's King from the novels: [Spoilers ASOS](# "- He was the 13th Lord Commander of the Night's Watch. - He fell in love with a woman 'with skin as white as the moon and eyes like blue stars.' '[The woman's] skin was cold as ice.' - 'When he gave his seed to her he gave his soul as well.' - He brought the woman back to the Nightfort, declared himself king, and her his queen. He continued to rule the Nightfort for 13 years. - After the King in the North and the King-beyond-the-Wall joined forces, they were able to defeat him. - It was later discovered that the Night's King, whose name was purged from all records and forgotten (as well as forbidden), was making sacrifices to the Others. - It is believed that the Night's Watch from that point forward no longer built walls around their castles to prevent another similar situation. - Some have said he was a Stark of Winterfell.").
[Spoilers Speculation based on the above plus a comment from one of Crastor's wives in ASOS](# "While he was stated as being human, there's some ambiguity in the story about what the Night's King became. The description of the woman he fell in love with matches that of the Others. Him 'losing his soul' when he had sex with her could be taken in a few different ways. It's possible he died and was reanimated as a wight. He may have been turned into an Other somehow. This concept is partially reinforced but not confirmed by a later comment in ASOS by one of Crastor's wives. Or it may have simply been a metaphorical loss of a soul since it was later discovered that he was making sacrifices to the Others. One could argue that doing such a thing would be inhuman.").
Additionally, GRRM has commented on the Night's King on his blog: [Spoilers GRRM](# "As for the Night’s King (the form I prefer), in the books he is a legendary figure, akin to Lann the Clever and Brandon the Builder, and no more likely to have survived to the present day than they have.").