Style guide question! Capitalizing "The Place Name"

If a place name starts with "The" and is used in the middle of a sentence, should "the" be capitalized in that spot? It's very common in bar names. I've tried consulting style guides and can't find it.

Examples:
He should have left them at T/the Prancing Pony, as they were clearly a bunch of food-obsessed tossers.
I'm going to T/the Cuff Friday night, while my girlfriend is having drinks at T/the Wild Rose.
Is T/the Wilde Rover actually queer-friendly, or do they just like putting a picture of Oscar Wilde on their sign?