If I were receiving an e-mail with attachments, I would want to be able to individually attest to the integrity of both “the message” and “each attachment.”   Furthermore, I would want to be able to vouch for each attachment even after it had been detached from the message.

So, there really are two separate problems here.   Standard S/MIME techniques are fine for the message, but I think that you need to view the issue for attachments as being more long-lasting.   Methinks that a digitally-signed message needs to be accompanied by (separately...) digitally-signed files as attachments, such that the authenticity of those files can be verified even weeks later, no matter what has been done to them in the meantime.   That takes you out of the intended scope of S/MIME, AFAIK.