Hi SNG
Yes, I think it's worth "casting the net" further and posting on the Canadian board particularly (and possibly also on the Australian board).

[I did search (some) Australian death and cemetery records - to no avail ... but much better to utilise the expertise of those with additional resources in Australia. ]
Incidentally, I see that Arthur was still in the UK (London area) in 1911 (boarding with a couple). It's possible that he worked his passage to NZ and that's why we've not found him amongst the passenger lists.
Good luck.

~ Lu