Is an absent BC and/or DC possible? The answer to both questions is yes.
A few years ago I did a project for a social historian/demographer - extracted birth notices from Wellington newspapers for 1875-1880; cross-referenced them with the birth registrations; also extracted relevant death notices (children <12 months of age) and cross-referenced those too.
I no longer have the exact figures (and I know it's a little earlier than the time period you want) BUT approx. 10-12% of annual births were not registered. This was especially so when the child died in the first 10-12 weeks. At this point there would often be a death registration but there was no earlier birth registration if the child had been ill since birth. Equally, there was no guarantee of either!
These days (and for the last 60 years or so) the medical person attending a birth has had a form to complete following a birth; the form is sent to the Central BDM Registry and later matched with the birth registration. Probably done online now but there are still gaps in the birth registration.
You may have to go back to the microfiche and extract relevant births from this source - as this will give you the place of registration; and then check this info against the BDM online. It's at this point you'll find out how much the BDM Central Registry leaves off the online indexes.