Well, since we have yet to see an authentic Barry LFBC or COLB, your guess is as good as mine.
I have for a long time assumed that Barry's birth certificate, when it eventually sees the light, will be based on an "at home birth" affidavit sworn by his Grandmother, Madelyn Dunham. This notion dovetails quite nicely with my current working theory as to what is the truth. It also fits the description of what Fukino said she saw in the records. She described it as "half written, half typed", which is pretty much what I would expect an "at home birth" affidavit to look like. Madelyn Dunham wrote out what happened on a piece of paper, and the registrar typed it out on that same paper, signed and stamped it.
Call me nuts, but this theory fits all the available information of which I am aware, and makes sense besides. Till a piece of information surfaces which makes it implausible, this is what i'm going with.