Don't citizens have to have an english-alphabet transliteration of their name to use for legal purposes (birth certificate, green card, social security record)?

Not in the US. In Japan and some nordic countries, only established names can be registered.

The DMV differentiates same-name people by SSN, right? Is it very far-fetched to imagine that state courts and federal appeals courts will uphold state laws requiring SSN disclosure for identification purposes?

I'm sorry, I only have a memory capacity of 7 plus or minus 2. I can deny my memory, particulary under stress of interrogation. Or I could use J. Edgar Hoovers which I sometimes mistake for mine.