Each manual typewriter has its own indivdual characteristics. They’d have to find one of the typewriters in use in the HI DoH at that time, not just the same make or style, but one of the exact ones that was in use.
Plus every file, archive, ledger and so on would have to be “fixed”.
Not very likely.
2) You could recreate documents that experts will swear were created not only on the same typewriter but by the same hand at the keys (based on impression depths) twenty years ago so getting beyond that level of detail is probably doable these days as well.
3) you seem to think that there would be unlimited right of inquiry into the document itself. If it passes a few obvious tests there would be no further testing, that's for sure.
I understand what you're saying about a trail of evidence and so forth, the problem is that a forgery that passes the most obvious tests is probably going to be accepted as valid thereby halting further study of said document. How everyone is going to rifle through sufficient files and ledgers to determine validity is something of a mystery to me given the fact that to date no one can even get a copy of any of the things Barry wants to keep hidden. Once he “gives in” and produces a BC that passes the initial obvious tests, there won't be any more digging. IOW, the tests that just might discern a forgery from the original would never be done if the document passes the most common battery of tests.
I honestly don't see the point anyway. Barry can have a full BC in his pocket and he still doesn't meet the Constitutional requirements so the whole issue of the BC is moot IMHO, although the issue of his being eligible isn't. This needs to be framed around the facts which are that due to the facts of his birth situation, it doesn't matter where he personally was born. That's the issue being dodged and the whole document search thing is a wasted of effort either way it goes.