Lu, thank you very much for the Cook Islands GenWeb information. It is greatly appreciated. Some of the links there are broken but most still work so I'll see what I can find.
Beg, your information is intriguing. I had seen the WWI Reserves Index information for Alexander (yes, I'm pretty sure it's him too) but I had completely missed the C before his name. And even if I'd seen it, I might not have thought much about it. If you can find the names of the two children, that would be wonderful as it could lead to additional information about Alexander's, and Vera's, time in the Cook Islands. I'm also thinking that Alexander and his first wife may have had more children since she did not die until 1924, aged 37.