John,
From the MIC index, I see that Arthur's middle name is Walter. Working on that and the possible Camberwell connection, the best fit I can find is this:
Births Dec 1888 Camberwell 1d 887
WILLIS, Arthur Walter
1891 Census RG12/475 Folio 106 Page 41
43 Crewys Rd, Peckham
Arthur W Willis, age 2, only child of Arthur J and Ruth C Willis. Arthur senior is a schoolteacher.
1901 Census RG13/502 Folio 78 Page 7
266 Friern Rd, Camberwell (East Dulwich)
Now aged 12, with parents and four siblings
I am waiting for a book to come through to my local library, from Southwark Libraries, specifically on the subject of the Camberwell Artillery. I will keep you in mind when I am going through it.
Phil