i'm sure registration of a child is not a requirement, but optional.
and as for the DNA test , that's only used as a last resort
http://www.ukvisas.gov.uk/en/ecg/set...ldren#18165823
If it appears that an illegitimate child has been brought up as a child of the family, it will normally be appropriate to admit the child under paragraph 297(i)(f). The fact that the sponsor may not be aware that the child is not his natural child should not preclude entry clearance.
but like you said, they would want to see your passport, but would they have noticed, the date ?, or if you had lost or had it stolen. maybe the birth cert would have been enough, but far better if you was married to her.
have you had legal advice yet ? your local law centre and
http://www.iasuk.org/home.aspx
i would have thought first thing to do would be getting your daughter her passport, seek legal advice from the above, and contact your MP, i've never heard anything so stupid, that you've had a DNA test and proved its your daughter, they are refusing to give her a passport.
wait for your g/f annulment to come thru, wed in the phils and apply for a settlement visa for your wife and son, you would be legally his stepfather, so really no need to adopt him, the only thing is you mention that your g/f has left her kids in the phils? to work in Korea, be careful of 'sole responsibility' that your g/f needs to prove with regards to her kids.
also if your g/f can come here for 6 months, is it not possible she can extend her stay, apply for a student visa? it would cost you money, but if you lived together and if no annulment yet, after 2yrs maybe you could apply for a unmarried partner visa.
or even if you could go and live in a european country exercise your treaty rights ( employed or self employed) after 3 months see if you can apply for a family permit for your g/f as a 'extended family member' then take her back to that european country with you,as different countries interpret the meaning of a 'durable relationship' differently (uk i think its 2yrs living together)
just waffling on, you need legal advice which is the best way forward for you..