I see a major problem.I don't think most royal houses would be interested in marrying their daughters over to the US if they can help it.Yup.
Actually, the Royal Marriages Act of OTL was the closest thing to a prohibition of morganatic marriages in the British royal family.
As to the marriage of the Princess Royal (if in a similar situation to Charlotte of Wales/Victoria) she'll marry a younger son of a foreign monarch. If she is never heiress apparent and the throne ends up coming to her out of the clouds (a la Electress Sophia/Queen Anne) she'll either be forced to relocate to America and marry someone of the Parliament's choosing, or if already married, nominate a second/younger son to serve as regent until she dies (if she doesn't/can't move across the pond), whereupon the younger son succeeds (sort of like Prince Alfred to Coburg or Ferdinand I to Romania). However, if sonny refuses to take up residence over the Atlantic, then they move on to the next heir (think of it sorta like the Act of Settlement's antipapist clause - you wanna be king of the USA you need to live here, not interested? Next please!)
As to the king-consort remarrying see Ferdinand II of Portugal remarrying to Else forget her last name, or Maria Cristina de Borbon, Queen of Spain's remarrying to Munoz or Caroline of Naples, duchesse de Berri to Prince Lucchesi-Palli. Their kids, despite being related to the royal family (Isabel II, Henri V or Pedro V), have no claims on the throne.