Beafan - lack of birth certificate is not a big deal (you can guess on the age...as long as your cat is more than 5 months old it's OK), but your cat MUST have the required shots (rabies, Feline Panleucopaenia (Infectious Enteritis) and Feline Respiratory Disease Complex (Cat Flu)).
Even if your local town doesn't have a vet you should be able to transport your cat to another town that does and then bring him/her into HK after that. I realize it's not easy, but if you really want your cat with you, it's possible.
NOTE/ Mainland China is considered a Group III country so there's a 4-month quarantine period in HK.