Capitalizing “The” in styles of nobility

Solution 1:

If you Google "the queen" and go to the results under "news" you'll see that most - including this, from the upmarket and monarchist Daily Telegraph - call her "the Queen".

Similarly, Googling "the duke of edinburgh", there are countless examples of TV stations and newspapers referring to him as "the Duke of Edinburgh".

Perhaps more importantly, the character in question wouldn't say "I am THE Lady Jane Doe". She would say, "I am Lady Jane Doe". She WOULD refer to her father as "the Duke of Utopia", because that is his title, not his name.

If the author really wants to say it I would recommend you leave "the" in lower case, despite what it says at that link.