Sorry to say this quite so bluntly, but even in the Edwardian days, when one partner died, there was nothing to stop a remaining partner like Ernestine going on to have another relationship without going through the formalities of a marriage. Instead of openly stating any offspring of that relationship was theirs however, a grandparent generally accepted full liability and brought their grandchild up pretending the child was actually theirs.
I agree that nothing is certain until you see the birth certificate, but if there is nothing on record circa 1892 under the name of Gertrude Smythe, the one for Gertrude Hart (Mother Ernestine?) seems a probable solution. You may see it as the loss of a certificate fee, but on the other hand, if it proves positive, your question will be answered.
Roy G