Q. Current guidelines on French capitalization in CMOS are mostly directed at an all-French context. In the case of an English text with a heavy dose of French proper names, it feels a bit awkward to leave the first word in a name such as “théâtre de la Porte Saint-Martin” with a lowercase t. What does the University of Chicago Press do in these cases?
A. The advice in Chicago is supposed to highlight conventions that can be retained when importing bits of French into an English-language context, and capitalization is one of these conventions. But an initial article like le or la is most often changed to an English the, so to use your example, we’d refer to the Théâtre de la Porte Saint-Martin (starting the name with a capital T) even if, in a French context, it might be referred to as le théâtre de la Porte Saint-Martin (the theater’s website, however, capitalizes that first t).
To take another example, you might refer in English to the Bibliothèque nationale de France or, for short, the Bibliothèque nationale, both of which reflect French capitalization (which is evident in the abbreviation BnF, with a lowercase n). That would normally be Chicago style—that is, we’d retain the lowercase n in English. But it would arguably be clearer to refer to the Bibliothèque Nationale, with a capital N, so that readers unfamiliar with French capitalization would understand where the name begins and ends. Some editors depart from Chicago style for such terms (and apply English-style caps, a.k.a. headline style or title case) for this very reason.
Note, however, that for the title of a book or article or other work, which would normally be set off from the surrounding text by quotation marks or italics, French capitalization would always be retained (see CMOS 11.27). But for the name of a theater or a library or the like, you could make some exceptions for the sake of your readers.