Is the T in "the" capitalized when referring to a royal house mid-sentence?

Every example I've been able to find has it at the beginning of the sentence, but while it seems like, unless it's actually part of a book title or company name or something, most proper nouns don't call for capitalizing the T in a preceding "the" mid-sentence, like people write "...the Beatles," not "...The Beatles," "...the East Coast," not "...The East Coast," and "...the President of the United States," not "...The President of The United States," I've seen that when Queen Elizabeth is referred to as a... I guess... pronoun of sorts, it's written "Her Majesty The Queen," the T in "the" getting capitalized. So I'm wondering if that's true of a royal family name, too, any royal family, not just the British one, for example:

The current British Royal Family is of The House of Windsor.

Or if that's not correct:

The current British Royal Family is of the House of Windsor.

So what's the proper way to write a royal family's name or house that's preceded by a "the" when it appears mid-sentence rather than at the beginning of a sentence, like was Tsar Nicholas II of the House of Romanov or of The House of Romanov?

The reason I'm asking is I think it is supposed to be capitalized, like I think I remember seeing that, but I'm not sure and I can't find anything that says, so maybe I'm just imagining it. I don't now.

According to the Oxford Learner’s Dictionary, the answer is no, you shouldn’t capitalise the article for a royal house:

the House of Windsor
/ðə ˌhaʊs əv ˈwɪnzə(r)/
/ðə ˌhaʊs əv ˈwɪnzər/
​ the name of the British royal family since 1917 when it was changed from Saxe-Coburg-Gotha

The Oxford English Dictionary also quotes the House of Windsor with an uncapitalised ‘the’, and the same for the House of Orange:

2002 Daily Mirror 31 May 15/1 A mish-mash of republican twaddle..seriously suggesting that we'd had enough of the House of Windsor.

1751 H. Walpole Lett. (1846) II. 404 It is become the peculiarity of the House of Orange to have minorities.

1848 W. K. Kelly tr. L. Blanc Hist. Ten Years I. 325 What are these treaties?.. Those of 1814? But these assure the possession of Belgium to the house of Orange.

1987 Fortune (Electronic ed.) 12 Oct. Like the House of Windsor, the House of Orange enjoys tax exemption, rendering anything more than an estimate of its wealth virtually impossible.

2002 Jakarta Post (Electronic ed.) 3 Aug. The orange tree is the symbol of the Royal House of Orange in the Netherlands.

It would appear that based on usage, the answer is that you should follow the same rule as with bands, newspapers and other proper nouns starting with an article: The determiner should not be capitalised unless it is the first letter of a full sentence, or if it is part of royal or religious address, as mentioned by @Greybeard in the comment below.