How can I update client_modified for a file through the API?

This is technically possible, but there are a few complicating factors:

• The only way to update client_modified is to perform an upload of the file with client_modified set.
• If you upload the same exact file data to an existing file, Dropbox won't consider this a change (since the file data didn't change) and so won't commit new metadata (such as an updated client_modified).

That being the case, you can make these changes, but you'll need to do so like you described, that is, by performing the upload with client_modified set, but as a new or changed file. (I'll send this along as a feature request for an easier/more efficient way to do this, but I can't promise if or when that might be implemented.)