I think the pink pregnancy book is exclusive for KKM use
Usually private clinic will provide their own record book (if they have the facility for accepting birth, if normal clinic one nope)

Its ok for bringing to private first then gomen, as gomen kk will provide one later.
Not sure what is the issue for the previous post above saying that the kk did not allow him/her to open one

as mentioned by tomato above

btw for my place during afternoon (aftter lunch break) not many people come

also, please notify your kampung kk later too (if your wife balik kampung after birth)