very interesting....... Despite seeming very obvious that the wording refers to a British Board of Trade Registration No., it's an idea that almost certainly is a non-starter. A four digit No. of this similarity would have been Registered in early March 1884, a few months after the use of lozenges had been abandoned permanently, and as James has commented there is nothing - at least in the reliable sources of Registrations - showing such a number - for CLASS III glass for anyone.
The nearest Davidson Registration I can see off-hand occurs in March 1888, and is a five digit number.
This is either from a mould of another factory, or might it be a George Davidson factory pattern design number? - but doesn't look to have anything to do with Board of Trade Registrations.