<blushing>
Yeah, that worked. Got a ton of logs, for both CAC and NEO. Will analyze and post here.
One weird thing - with CAC, even though it can't be unlocked by Keychain Access (and its certs don't seem visible by Apple Mail), I could successfully configure MS Outlook 2011 to use CAC to sign email (verified - it worked). But Keychain saw and reported on the private keys as well, just couldn't unlock.
With NEO - Keychain does not see the private keys at all, only the certs. And no other tokend-related app (that I tried) was able to do anything with NEO PIV. (So far, that is.)