I see. May I address your following points:
-This Prince Hubert was a immediate member of the royal family, being one of the sons of Henry IV. Since you said that name was used amongst royal family members, it would count.
-Elizabeth of Northumberland is related to the King.
-Is it necessary to rely on the authority of a foreign power in order to create a kingly title?
-The unification engineered by Henry I (or VII in our timeline and before the unification in mines) is a real union, unlike the personal union that had existed previously from the time of Edward III. As you would note, England and Scotland remained separate kingdoms, with their own governments, militaries, and laws, until Henry I proclaimed himself Emperor.
-Henry VIII (I) would have been dis-satisfied with Katherine of Aragon because of her inability to bear children, and would claim that she had slept with his (in this case younger) brother Arthur before marrying him, thus making the marriage unlawful.
-Henry I (VII) would have adopted the new regnal numberings because he believed Brittania was the successor state (not a continuation) of the Kingdom of England, the Kingdom of Scotland, and the Lordship of Ireland, with all the previous rulers being "rulers of predecessor states".