You forget, it's illegal for them to release the information on "vital records", which include Certificates of Live Birth, Certifications too, except to a very limited number of persons. Absent a court order/subpoena that is.
They can't complain.
No, I didn't forget that, but this is the way I see it:
1. If I were a high official of the Hawaiian DOH and I saw that a presidential candidate was presenting a forged document as if it were issued by my department, and that history might turn on this constitutionally-relevant issue, I would obey my oath and the law, but I would also try to think of some way of aiding others in discovering the fact of the forgery.
For example, if the father's race on a birth record in 1961 could only be selected from a limited number of choices (which did not include "African"), I could make sure that the DOH was asked about this.
2. If Fukino can publicly state that Obama was born in Honolulu and claim that therefore he is is a natural born citizen, she could certainly have gone so far as to point out a forgery.
3. Of course it is possible, as you say, that even knowing of a serious constitutional issue, the DOH officials are bureaucratically following the statute 338-18, which does in fact state that they are not even supposed to verify a vital record except in certain cases such as a court order. But I think #2 argues against that.
BTW this law seems to say that exceptions can be made according to Department of Health rules.