What gives me some hope is that this is not a birth certificate. It is a "Certified Copy of Registration of Birth."
It gives a link to a page in a hard copy book (i.e., a register) that would contain the information quoted on the form we see.
For example, here is a page from an 1883 UK birth register:
I cannot picture that a 1961 register would be any different (computerization would not have come to an outlying colony in 1961 -- particularly one that the Brits knew they were going to be quitting in a couple of short years anyway)
The point is excising one entry from such a register would be impossible. Excising a whole page from such a register would be near to impossible. Excising a whole register from a set of registers would be really, really suspicious (and difficult to do, to boot).