My honest opinion is that there’s SO much occult and spiritual power wrapped up in that office (of royalty) that you don’t stand a gnat in hell’s chance to work magick ON them, you may be able to wangle circs where one guy dies and you marry his widow and whatever, but that won’t make you a prince, and short of literal reality-editing I don’t think plain evocation is going to give you a route to this.
I don’t think writing a letter like a guy pleading is going to help, you’re kind of asking for godlike power to alter reality and yet coming at it like someone asking for a request song to be played on a radio station, the eternal outsider, who has to hope and watch and ask from the other side of the mailbox.
If you wanted to infiltrate Royal circles and be in with a chance to marry maybe the bun Kate has in her oven (assuming it’s female) that would be viable, so would any other approach using existing power structures and relationships.
But I think the chance of your letter even reaching the Queen’s own assistant is pretty slim, and more likely to get you on a watch list for people with a pathological need to be inside the family than anything else (not being rude to your goals there, mine are madder, just trying to help).
People who want to be in someone’s life, inside a specific social circle, and enjoying their good things without having earned it can get very pissy when rejected, and security know this - go watch The Talented Mr Ripley to see an example of an outsider and how he acts when he’s let in amongst the beautiful people (Weinstein be damned, that’s a fun movie).