Family mystery

My granddfather was one of 12 children, of whom 9 survived to adulthood. Two brothers went to Canada at different times, and a sister later on, and I am in contact with their families. One "cousin" is a regular correspondent by e-mail and is trying to piece together a family history, with the help of her late grandfathers photo album. One of the brothers (Arthur) who stayed in the UK married before World War One and had one son. Then my cousin in Canada came across some photos of "Betty" in 1932, in which said Betty looks around 5 or 6 years old. One of the photos also describes her as the daughter of Arthur. Now Arthur died in 1917 and neither the info given to me by my Granddad, or information checked out with another cousin (grandchild of one of my Great Aunts) ever mentioned Arthur having more than one child. We are now coming to the conclusion that Betty may have been Arthur's widow's daughter by her second husband. So now I need to try and find Arthur's marriage details to establish his wife's maiden name and then see if I can find Betty by reference to the surname we think was Arthur's widow's second husband's name. But although I think I have found the entry in the BMD of Arthur's marriage, it is before that record recorded the name of the other person so that would mean ordering a certificate without knowing if it was the right one and it's all going to get too expensive. One of these days I am going to try to draw a family tree and I have come to the conclusion that I will have to do it on a roll of lining paper stuck to the bedroom wall!