Heh, the problem isn't really with the choice of label, it's that the thing it labels is really hard to pin down. Most attempts wind up meaningless or controversial.
But there is a colloquial usage, obviously. A radio station near me claims to play "Great Classical Music." Well, what do they not play? No rock and roll. No jazz. No hip hop. Generally, the most useful man-on-the-street sense of what makes something "classical" has a lot to do with the instruments used. Adding a violin to the band Yellowcard doesn't make them classical, and adding a drum set to a symphony orchestra for Rhapsody in Blue doesn't make it not-classical, but in general, using the instruments of the symphony orchestra tends to cause a perception of "classical."
But obviously this colloquial sense is of little use to scholarly discussion. Is a movie soundtrack "classical"? Many will vehemently deny it. The classical radio station is unlikely to play it. (But they will play music from ballets or operas...) Is a Vitamin C String Quartet cover of a pop song "classical"? Is John Cage's 4'33" (a piece consisting of silence) classical? Is Steve Reich's Electric Counterpoint, written for/with jazz guitarist Pat Metheny? What about when Johnny Greenwood, guitarist from Radiohead, plays it? What about when Steve Reich reworks Radiohead songs in Radio Rewrite?
If one's scholarship were sociological, it could be a rich (perhaps impossible) task to parse all these conflicting uses. But in the meantime, rather than find a different label, it's better to nail down exactly what one is talking about.
The term "art music" is marginally more useful, but still very problematic. It carries a lot of baggage. It's a step more generous than simply talking about "music," and then talking in a way that ignores non-classical genres. It admits that folk music and pop music are actually "music," but then it tries to exclude them, and the word it chooses for that purpose is "Art." There's already plenty of ink spilled over that word, clear back to the ancient Greeks, and we're implying that folk music and pop music are not "art." Functionally, the best this term can do is try to focus on the "use" and context of the music. Is it played in a concert hall, or an outdoor festival? Is it played to people who sit still and listen to it, or is does it contribute to another activity, like dance music in a club? Ultimately, this distinction will break down under scrutiny as well. A string quartet, a jazz combo, and an indie rock band might all play in medium-size venues, to attentive audiences who came specifically to hear them. The only distinctions become cultural "highbrow"/"lowbrow"/"middlebrow" tokens. We even open the door to the implication that "Art" is the opposite of "Entertainment," and that therefore "classical" equals "serious music" (as opposed to what? music that can smile?). I once saw, backstage in a concert hall, an old lighting control console from the 1920s, no longer in use. Someone had added joking labels to the various dials and levers. One was a big handle that pivoted vertically, and the top of the scale was labeled "Art" and the bottom "Entertainment." Phrases like "art music" lead inevitably to this lever—the more fun it is, the less artistic, and vice versa.
The best thing, then, is to narrow your topic of discussion. Are you talking about the social role of classical music institutions? It's easy enough to talk specifically about symphonies, opera companies, etc. Are you contrasting this music with that of another global music-culture? I often use "Western tonal tradition" or words to that effect to mean "the monolithic musical practice that derives mostly from European origin, starting roughly in the baroque, and that became codified by these social institutions like symphonies and continues to be practiced by them, and that is marked by musical conventions like tonality and regular meter." But if at all possible it's more fruitful to draw more specific comparisons. I once wrote a paper about how "classical music" came to be such a big deal to Chinese culture, but it's a story of individual connections and influences, of a clavichord in the Emperor's court and a symphony in Shanghai. I got most of it from the book Rhapsody in Red: How Western Classical Music Became Chinese by Sheila Melvin and Jindong Cai. (I highly recommend it, a great read.) Note that they, at least, pick the term "Western Classical," but they devote a paragraph to talking about the fact that they picked it.
Throughout the writing of this book, we have grappled with our use of the term "classical music." It is a phrase that we find to be deeply flawed and woefully inexact, but largely unavoidable. Because it had not yet been coined during most of the time period covered in Chapter 2, we have substituted "European music" instead. But in writing about the missionaries who used music as a tool of conversion and education in the 19th and early 20th centuries, we switch to "Western music" since many of them were American and the music they taught was not strictly-speaking "European" or "classical." However, as we got deeper into the 20th century, we were largely forced to drop "Western music" because by then the term could include jazz, pop, rock, disco, and rap, none of which we discuss. So, for the title of our book and through many of its chapters, we have been obliged to use the phrase "classical music" to refer to a broad range of music that was originally brought to China from the West and includes the clavichord, the military march, opera, the symphony, the piano concerto, and more. If the phrase and our use of it are far from perfect, we at least find comfort in the fact that even The Oxford Dictionary of Music defines classical music as a "term which, applied to music, has vague rather than specific meaning." (p.3)
And that's about the only way we can talk about these things—defining the scope of your conversation, and taking time to talk about the terms you've chosen, their inherent problems, and what you mean for them to mean.
Update: I feel like, after that, I'm lacking a clear answer to the original question, "If 'classical music' is an inadequate label, what term should I use?" My answer, then would be:
- First, try not to talk about such a monolithic and poorly-defined entity. Wherever possible, talk about more specific slices of history and culture. And,
- If you must talk about it (e.g. if you're talking about the sociological phenomenon of people perceiving a body of "classical music"), I don't know that any other label would avoid the same problems with definition and boundaries. Instead, do as Melvin and Cai did above; be open about its problems and clear about what meanings and boundaries you intend, and when you discuss others' use about it, make their meanings clear as well (or call out their unclearness).