While foreign names for The Hague are similar to the now more used Dutch name Den Haag, its other name still official in use ...
Thanks for the info - and I'm not being contrary here - but the official name is not the only name which might be validly treated as a proper noun for some person, place, or thing.
An official name definitely is a proper name and should always be capitalised. But - and this is a grey area - nicknames may also be well-enough established to be capitalised as proper nouns too. For example, should you capitalise both of these names of a famous golfer, Tiger Woods and Eldrick Woods? Obviously, you should.
The grey area is with pet names, names that one person frequently calls another. Style guide state they should not be capitalised, but there appears to be a consensus among those here that authors are, at the very least, entitled to treat some pet names as proper nouns.
SB cited an example of a father who calls his daughter "Princess". IMHO, if the father always calls her that or her actual name, that is sufficient to justify treating it as a nickname, but my interpretation of CMOS is that they do not agree. I think they consider a nickname is only something most people would recognise as an alternative name for a person.
Then again - as always - consistency trumps any rule. I would not do it, but I wouldn't argue an author was doing anything wrong if they used capitals for everything used as a substitute for someone's name.
Then again - I recently did precisely that myself! I was editing a story in which the author was using a lot of unusual pet names, things like connivance, destroyer, and a variety of less inflammatory ordinary words. I recommended, for just that story, they should capitalise all such usages, on the grounds that otherwise the risk was too great readers would become confused trying to interpret those words with their standard meaning.
So, going back to Den Haag ... I would still write that as 'The Hague' in all situations. It may only be a nickname, but it's one that everyone would recognise, and it's always(?) said with the word 'The'.
I just checked ngrams. When 'Hague' was used, 61% were preceded by 'The', 17% by 'the', and 23% by neither.