Hi Questor
Over the last few days, I have had a further look at this and tried various avenues and ideas (no doubt replicating what you have already done), but I cannot find any new information for you.
It is possible that the birth registration you have attached with mother as Elizabeth Bendall is for your Alfred, but the mother made up some names to cover an illegitimate birth. As you know there are various details that make this a likely match, but other details raise questions.
It is also possible that your Alfred is recorded under some other surname - I tried a few like mother's maiden name and her mother's name, etc.
It is probable that the father was named "Alfred" but you may never know his surname. Alfred considered that Hugh (his step-father) was his "father", but I don't believe he was the biological father.
My only other thought is that George Shaw had a number of brothers, and I wonder if one of the brothers was the real father to your Alfred.