isbndb has Dewey classification numbers for some titles, but I have no idea how comprehensive it is or whether the API is set up for that info to be collected. I do know that there are software applications for cataloging pbooks that include this functionality, but as I no longer have any of them (gave up because I didn't want to buy an ISBN scanner) I don't know where the information is being collected from.
As much as I dislike LOC classifications, at least with LOC there's a verifiable database already in place, though I don't know what the API details are for that either. I imagine the info is on the LOC website but I have a headache and my brain isn't up to it.
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