Authenticating with a document
Your idea seems secure, but it is more complicated than it needs to be.

If you only need to know if the "secure information" in the document matches some public information (stored on an insecure computer), it is enough to calculate a preimage resistant hash of the document (e.g. SHA-256). The document needs only to be unguessable, like a 256-bit random number.

Note that there are practical problems here, which may lead to side channel attacks. For example, doing anything on a general purpose computer usually involves a chance that the memory you are using hits the swap on the disk, unless you use something like mlock.

You would need to carefully consider which parts of the computer you consider secure and what kind of attacks may be possible.