I can't think of one epithet that would do for all the very different types of art music/ classical music.
'Elegant' is a word that I often use for baroque music, for example - but it can be a put-down when applied to works like Biber's Battalia or Handel's Messiah.
I think the point - for me - about classical music is that more than other types of music (generalising here) it shows evidence of 'mind'. 'Thoughtful' music, however, doesn't do the job, because it sounds too cautious for something like Beethoven's Fifth or the Ritual Fire Dance.
So I thought - 'Designer' music?
But that sounds too chic, music meant only for those listeners who have a 'lifestyle' (something that I've aspired to for years, in vain

).
'Art' music does the job as well as anything - it implies a mind, it implies beauty or other arresting qualities, and it implies an audience that wants something more than instant gratification. It also implies a certain timelessness.
Of course, it has its problems - people can't agree on the definition of art, and also non-classical music can have designer qualities, beauty, and timelessness too.
'Classical' music? The term seems to be all about timelessness, but is also blurred by the use of the word for Graeco-Roman Civilisation and the music of Mozart.
But although I don't think there's an answer, I do think that it is an interesting question worth asking.